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 The early search for stellar parallax: Galileo, Castelli, and RamponiNot Available Decay of Planetary Debris DisksWe report new Spitzer 24 μm photometry of 76 main-sequence A-typestars. We combine these results with previously reported Spitzer 24μm data and 24 and 25 μm photometry from the Infrared SpaceObservatory and the Infrared Astronomy Satellite. The result is a sampleof 266 stars with mass close to 2.5 Msolar, all detected toat least the ~7 σ level relative to their photospheric emission.We culled ages for the entire sample from the literature and/orestimated them using the H-R diagram and isochrones; they range from 5to 850 Myr. We identified excess thermal emission using an internallyderived K-24 (or 25) μm photospheric color and then compared allstars in the sample to that color. Because we have excluded stars withstrong emission lines or extended emission (associated with nearbyinterstellar gas), these excesses are likely to be generated by debrisdisks. Younger stars in the sample exhibit excess thermal emission morefrequently and with higher fractional excess than do the older stars.However, as many as 50% of the younger stars do not show excessemission. The decline in the magnitude of excess emission, for thosestars that show it, has a roughly t0/time dependence, witht0~150 Myr. If anything, stars in binary systems (includingAlgol-type stars) and λ Boo stars show less excess emission thanthe other members of the sample. Our results indicate that (1) there issubstantial variety among debris disks, including that a significantnumber of stars emerge from the protoplanetary stage of evolution withlittle remaining disk in the 10-60 AU region and (2) in addition, it islikely that much of the dust we detect is generated episodically bycollisions of large planetesimals during the planet accretion end game,and that individual events often dominate the radiometric properties ofa debris system. This latter behavior agrees generally with what we knowabout the evolution of the solar system, and also with theoreticalmodels of planetary system formation. CHARM2: An updated Catalog of High Angular Resolution MeasurementsWe present an update of the Catalog of High Angular ResolutionMeasurements (CHARM, Richichi & Percheron \cite{CHARM}, A&A,386, 492), which includes results available until July 2004. CHARM2 is acompilation of direct measurements by high angular resolution methods,as well as indirect estimates of stellar diameters. Its main goal is toprovide a reference list of sources which can be used for calibrationand verification observations with long-baseline optical and near-IRinterferometers. Single and binary stars are included, as are complexobjects from circumstellar shells to extragalactic sources. The presentupdate provides an increase of almost a factor of two over the previousedition. Additionally, it includes several corrections and improvements,as well as a cross-check with the valuable public release observationsof the ESO Very Large Telescope Interferometer (VLTI). A total of 8231entries for 3238 unique sources are now present in CHARM2. Thisrepresents an increase of a factor of 3.4 and 2.0, respectively, overthe contents of the previous version of CHARM.The catalog is only available in electronic form at the CDS viaanonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/431/773 Nearby stars of the Galactic disk and halo. III.High-resolution spectroscopic observations of about 150 nearby stars orstar systems are presented and discussed. The study of these and another100 objects of the previous papers of this series implies that theGalaxy became reality 13 or 14 Gyr ago with the implementation of amassive, rotationally-supported population of thick-disk stars. The veryhigh star formation rate in that phase gave rise to a rapid metalenrichment and an expulsion of gas in supernovae-driven Galactic winds,but was followed by a star formation gap for no less than three billionyears at the Sun's galactocentric distance. In a second phase, then, thethin disk - our familiar Milky Way'' - came on stage. Nowadays ittraces the bright side of the Galaxy, but it is also embedded in a hugecoffin of dead thick-disk stars that account for a large amount ofbaryonic dark matter. As opposed to this, cold-dark-matter-dominatedcosmologies that suggest a more gradual hierarchical buildup throughmergers of minor structures, though popular, are a poor description forthe Milky Way Galaxy - and by inference many other spirals as well - if,as the sample implies, the fossil records of its long-lived stars do notstick to this paradigm. Apart from this general picture that emergeswith reference to the entire sample stars, a good deal of the presentwork is however also concerned with detailed discussions of manyindividual objects. Among the most interesting we mention the bluestraggler or merger candidates HD 165401 and HD 137763/HD 137778, thelikely accretion of a giant planet or brown dwarf on 59 Vir in itsrecent history, and HD 63433 that proves to be a young solar analog at\tau200 Myr. Likewise, the secondary to HR 4867, formerly suspectednon-single from the Hipparcos astrometry, is directly detectable in thehigh-resolution spectroscopic tracings, whereas the visual binary \chiCet is instead at least triple, and presumably even quadruple. Withrespect to the nearby young stars a complete account of the Ursa MajorAssociation is presented, and we provide as well plain evidence foranother, the Hercules-Lyra Association'', the likely existence ofwhich was only realized in recent years. On account of its rotation,chemistry, and age we do confirm that the Sun is very typical among itsG-type neighbors; as to its kinematics, it appears however not unlikelythat the Sun's known low peculiar space velocity could indeed be thecause for the weak paleontological record of mass extinctions and majorimpact events on our parent planet during the most recent Galactic planepassage of the solar system. Although the significance of thiscorrelation certainly remains a matter of debate for years to come, wepoint in this context to the principal importance of the thick disk fora complete census with respect to the local surface and volumedensities. Other important effects that can be ascribed to this darkstellar population comprise (i) the observed plateau in the shape of theluminosity function of the local FGK stars, (ii) a small thoughsystematic effect on the basic solar motion, (iii) a reassessment of theterm asymmetrical drift velocity'' for the remainder (i.e. the thindisk) of the stellar objects, (iv) its ability to account for the bulkof the recently discovered high-velocity blue white dwarfs, (v) itsmajor contribution to the Sun's 220 km s-1 rotationalvelocity around the Galactic center, and (vi) the significant flatteningthat it imposes on the Milky Way's rotation curve. Finally we note ahigh multiplicity fraction in the small but volume-complete local sampleof stars of this ancient population. This in turn is highly suggestivefor a star formation scenario wherein the few existing single stellarobjects might only arise from either late mergers or the dynamicalejection of former triple or higher level star systems. Some anomalies in the occurrence of debris discs around main-sequence A and G starsDebris discs consist of large dust grains that are generated bycollisions of comets or asteroids around main-sequence stars, and thequantity and distribution of debris may be used to detect the presenceof perturbing planets akin to Neptune. We use stellar and disc surveysto compare the material seen around A- and G-type main-sequence stars.Debris is detected much more commonly towards A stars, even when acomparison is made only with G stars of comparable age. Detection ratesare consistent with disc durations of ~0.5 Gyr, which may occur at anytime during the main sequence. The higher detection rate for A stars canresult from this duration being a larger fraction of the main-sequencelifetime, possibly boosted by a globally slightly larger disc mass thanfor the G-type counterparts. The disc mass range at any given age is afactor of at least ~100 and any systematic decline with time is slow,with a power law estimated to not be steeper than t-1/2.Comparison with models shows that dust can be expected as late as a fewGyr when perturbing planetesimals form slowly at large orbital radii.Currently, the Solar system has little dust because the radius of theKuiper Belt is small and hence the time-scale to produce planetesimalswas less than 1 Gyr. However, the apparently constant duration of ~0.5Gyr when dust is visible is not predicted by the models. Contributions to the Nearby Stars (NStars) Project: Spectroscopy of Stars Earlier than M0 within 40 Parsecs: The Northern Sample. I.We have embarked on a project, under the aegis of the Nearby Stars(NStars)/Space Interferometry Mission Preparatory Science Program, toobtain spectra, spectral types, and, where feasible, basic physicalparameters for the 3600 dwarf and giant stars earlier than M0 within 40pc of the Sun. In this paper, we report on the results of this projectfor the first 664 stars in the northern hemisphere. These resultsinclude precise, homogeneous spectral types, basic physical parameters(including the effective temperature, surface gravity, and overallmetallicity [M/H]), and measures of the chromospheric activity of ourprogram stars. Observed and derived data presented in this paper arealso available on the project's Web site. Stellar Kinematic Groups. II. A Reexamination of the Membership, Activity, and Age of the Ursa Major GroupUtilizing Hipparcos parallaxes, original radial velocities and recentliterature values, new Ca II H and K emission measurements,literature-based abundance estimates, and updated photometry (includingrecent resolved measurements of close doubles), we revisit the UrsaMajor moving group membership status of some 220 stars to produce afinal clean list of nearly 60 assured members, based on kinematic andphotometric criteria. Scatter in the velocity dispersions and H-Rdiagram is correlated with trial activity-based membership assignments,indicating the usefulness of criteria based on photometric andchromospheric emission to examine membership. Closer inspection,however, shows that activity is considerably more robust at excludingmembership, failing to do so only for <=15% of objects, perhapsconsiderably less. Our UMa members demonstrate nonzero vertex deviationin the Bottlinger diagram, behavior seen in older and recent studies ofnearby young disk stars and perhaps related to Galactic spiralstructure. Comparison of isochrones and our final UMa group membersindicates an age of 500+/-100 Myr, some 200 Myr older than thecanonically quoted UMa age. Our UMa kinematic/photometric members' meanchromospheric emission levels, rotational velocities, and scattertherein are indistinguishable from values in the Hyades and smaller thanthose evinced by members of the younger Pleiades and M34 clusters,suggesting these characteristics decline rapidly with age over 200-500Myr. None of our UMa members demonstrate inordinately low absolutevalues of chromospheric emission, but several may show residual fluxes afactor of >=2 below a Hyades-defined lower envelope. If one defines aMaunder-like minimum in a relative sense, then the UMa results maysuggest that solar-type stars spend 10% of their entire main-sequencelives in periods of precipitously low activity, which is consistent withestimates from older field stars. As related asides, we note six evolvedstars (among our UMa nonmembers) with distinctive kinematics that liealong a 2 Gyr isochrone and appear to be late-type counterparts to diskF stars defining intermediate-age star streams in previous studies,identify a small number of potentially very young but isolated fieldstars, note that active stars (whether UMa members or not) in our samplelie very close to the solar composition zero-age main sequence, unlikeHipparcos-based positions in the H-R diagram of Pleiades dwarfs, andargue that some extant transformations of activity indices are notadequate for cool dwarfs, for which Ca II infrared triplet emissionseems to be a better proxy than Hα-based values for Ca II H and Kindices. STELIB: A library of stellar spectra at R ~ 2000We present STELIB, a new spectroscopic stellar library, available athttp://webast.ast.obs-mip.fr/stelib. STELIB consists of an homogeneouslibrary of 249 stellar spectra in the visible range (3200 to 9500Å), with an intermediate spectral resolution (la 3 Å) andsampling (1 Å). This library includes stars of various spectraltypes and luminosity classes, spanning a relatively wide range inmetallicity. The spectral resolution, wavelength and spectral typecoverage of this library represents a substantial improvement overprevious libraries used in population synthesis models. The overallabsolute photometric uncertainty is 3%.Based on observations collected with the Jacobus Kaptein Telescope,(owned and operated jointly by the Particle Physics and AstronomyResearch Council of the UK, The Nederlandse Organisatie voorWetenschappelijk Onderzoek of The Netherlands and the Instituto deAstrofísica de Canarias of Spain and located in the SpanishObservatorio del Roque de Los Muchachos on La Palma which is operated bythe Instituto de AstrofÃsica de Canarias), the 2.3 mtelescope of the Australian National University at Siding Spring,Australia, and the VLT-UT1 Antu Telescope (ESO).Tables \ref{cat1} to \ref{cat6} and \ref{antab1} to A.7 are onlyavailable in electronic form at http://www.edpsciences.org. The StellarLibrary STELIB library is also available at the CDS, via anonymous ftpto cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/402/433 Empirical calibration of the near-infrared Ca II triplet - III. Fitting functionsUsing a near-infrared stellar library of 706 stars with a wide coverageof atmospheric parameters, we study the behaviour of the CaII tripletstrength in terms of effective temperature, surface gravity andmetallicity. Empirical fitting functions for recently definedline-strength indices, namely CaT*, CaT and PaT, are provided. Thesefunctions can be easily implemented into stellar population models toprovide accurate predictions for integrated CaII strengths. We alsopresent a thorough study of the various error sources and their relationto the residuals of the derived fitting functions. Finally, the derivedfunctional forms and the behaviour of the predicted CaII are comparedwith those of previous works in the field. Metallicity Determinations from Ultraviolet-Visual Spectrophotometry. I. The Test SampleNew visual spectrophotometric observations of non-supergiant solarneighborhood stars are combined with IUE Newly Extracted Spectra (INES)energy distributions in order to derive their overall metallicities,[M/H]. This fundamental parameter, together with effective temperatureand apparent angular diameter, is obtained by applying the flux-fittingmethod while surface gravity is derived from the comparison withevolutionary tracks in the theoretical H-R diagram. Trigonometricparallaxes for the stars of the sample are taken from the HipparcosCatalogue. The quality of the flux calibration is discussed by analyzinga test sample via comparison with external photometry. The validity ofthe method in providing accurate metallicities is tested on a selectedsample of G-type stars with well-determined atmospheric parameters fromrecent high-resolution spectral analysis. The extension of the overallprocedure to the determination of the chemical composition of all theINES non-supergiant G-type stars with accurate parallaxes is planned inorder to investigate their atmospheric temperature structure. Based onobservations collected at the INAOE G. Haro'' Observatory, Cananea(Mexico). Rotational velocities of A-type stars in the northern hemisphere. II. Measurement of v sin iThis work is the second part of the set of measurements of v sin i forA-type stars, begun by Royer et al. (\cite{Ror_02a}). Spectra of 249 B8to F2-type stars brighter than V=7 have been collected at Observatoirede Haute-Provence (OHP). Fourier transforms of several line profiles inthe range 4200-4600 Å are used to derive v sin i from thefrequency of the first zero. Statistical analysis of the sampleindicates that measurement error mainly depends on v sin i and thisrelative error of the rotational velocity is found to be about 5% onaverage. The systematic shift with respect to standard values fromSlettebak et al. (\cite{Slk_75}), previously found in the first paper,is here confirmed. Comparisons with data from the literature agree withour findings: v sin i values from Slettebak et al. are underestimatedand the relation between both scales follows a linear law ensuremath vsin inew = 1.03 v sin iold+7.7. Finally, thesedata are combined with those from the previous paper (Royer et al.\cite{Ror_02a}), together with the catalogue of Abt & Morrell(\cite{AbtMol95}). The resulting sample includes some 2150 stars withhomogenized rotational velocities. Based on observations made atObservatoire de Haute Provence (CNRS), France. Tables \ref{results} and\ref{merging} are only available in electronic form at the CDS viaanonymous ftp to cdsarc.u-strasbg.fr (130.79.125.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/393/897 CHARM: A Catalog of High Angular Resolution MeasurementsThe Catalog of High Angular Resolution Measurements (CHARM) includesmost of the measurements obtained by the techniques of lunaroccultations and long-baseline interferometry at visual and infraredwavelengths, which have appeared in the literature or have otherwisebeen made public until mid-2001. A total of 2432 measurements of 1625sources are included, along with extensive auxiliary information. Inparticular, visual and infrared photometry is included for almost allthe sources. This has been partly extracted from currently availablecatalogs, and partly obtained specifically for CHARM. The main aim is toprovide a compilation of sources which could be used as calibrators orfor science verification purposes by the new generation of largeground-based facilities such as the ESO Very Large Interferometer andthe Keck Interferometer. The Catalog is available in electronic form atthe CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/386/492, and from theauthors on CD-Rom. Astrometric radial velocities. III. Hipparcos measurements of nearby star clusters and associationsRadial motions of stars in nearby moving clusters are determined fromaccurate proper motions and trigonometric parallaxes, without any use ofspectroscopy. Assuming that cluster members share the same velocityvector (apart from a random dispersion), we apply a maximum-likelihoodmethod on astrometric data from Hipparcos to compute radial and spacevelocities (and their dispersions) in the Ursa Major, Hyades, ComaBerenices, Pleiades, and Praesepe clusters, and for theScorpius-Centaurus, alpha Persei, and HIP 98321'' associations. Theradial motion of the Hyades cluster is determined to within 0.4 kms-1 (standard error), and that of its individual stars towithin 0.6 km s-1. For other clusters, Hipparcos data yieldastrometric radial velocities with typical accuracies of a few kms-1. A comparison of these astrometric values withspectroscopic radial velocities in the literature shows a good generalagreement and, in the case of the best-determined Hyades cluster, alsopermits searches for subtle astrophysical differences, such as evidencefor enhanced convective blueshifts of F-dwarf spectra, and decreasedgravitational redshifts in giants. Similar comparisons for the ScorpiusOB2 complex indicate some expansion of its associations, albeit slowerthan expected from their ages. As a by-product from the radial-velocitysolutions, kinematically improved parallaxes for individual stars areobtained, enabling Hertzsprung-Russell diagrams with unprecedentedaccuracy in luminosity. For the Hyades (parallax accuracy 0.3 mas), itsmain sequence resembles a thin line, possibly with wiggles in it.Although this main sequence has underpopulated regions at certaincolours (previously suggested to be Böhm-Vitense gaps''), suchare not visible for other clusters, and are probably spurious. Futurespace astrometry missions carry a great potential for absoluteradial-velocity determinations, insensitive to the complexities ofstellar spectra. Based on observations by the ESA Hipparcos satellite.Extended versions of Tables \ref{tab1} and \ref{tab2} are available inelectronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr(130.79.125.8) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/381/446 The Physical Basis of Luminosity Classification in the Late A-, F-, and Early G-Type Stars. I. Precise Spectral Types for 372 StarsThis is the first in a series of two papers that address the problem ofthe physical nature of luminosity classification in the late A-, F-, andearly G-type stars. In this paper, we present precise spectralclassifications of 372 stars on the MK system. For those stars in theset with Strömgren uvbyβ photometry, we derive reddenings andpresent a calibration of MK temperature types in terms of the intrinsicStrömgren (b-y)0 index. We also examine the relationshipbetween the luminosity class and the Strömgren c1 index,which measures the Balmer jump. The second paper will address thederivation of the physical parameters of these stars, and therelationships between these physical parameters and the luminosityclass. Stars classified in this paper include one new λ Bootisstar and 10 of the F- and G-type dwarfs with recently discoveredplanets. The geometry of stellar motions in the nucleus region of the Ursa Major kinematic groupThe structure of the central part of the UMa stellar kinematic group wasconsidered according to the data from Hipparcos, Tycho and ACTcatalogues. We have shown that in the space occupied by the nucleusthere are several populations of stars. The first includes ten starswhich are very compactly located in the proper motion diagram and have anormal proper motion distribution. The second group includes six starswith large kinematic difference and there are six field stars placedhere too. The question about their coexistence in a common region ofspace remains open. The well-known stars Mizar and Alcor appear tobelong to the different subsystems and do not belong to a unique stellarsystem. The published radial velocities, as a rule, are determined withinsufficient accuracy compared to the errors of the currently availableproper motions. A high precision in the radial velocities measurementsis necessary for a thorough study of the kinematics of stars inside UMastar system. Catalogue of Apparent Diameters and Absolute Radii of Stars (CADARS) - Third edition - Comments and statisticsThe Catalogue, available at the Centre de Données Stellaires deStrasbourg, consists of 13 573 records concerning the results obtainedfrom different methods for 7778 stars, reported in the literature. Thefollowing data are listed for each star: identifications, apparentmagnitude, spectral type, apparent diameter in arcsec, absolute radiusin solar units, method of determination, reference, remarks. Commentsand statistics obtained from CADARS are given. The Catalogue isavailable in electronic form at the CDS via anonymous ftp tocdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcar?J/A+A/367/521 Incidence and survival of remnant disks around main-sequence starsWe present photometric ISO 60 and 170 μm measurements, complementedby some IRAS data at 60 μm, of a sample of 84 nearby main-sequencestars of spectral class A, F, G and K in order to determine theincidence of dust disks around such main-sequence stars. Fifty starswere detected at 60 μm. 36 of these emit a flux expected from theirphotosphere while 14 emit significantly more. The excess emission weattribute to a circumstellar disk like the ones around Vega and betaPictoris. Thirty four stars were not detected at all; the expectedphotospheric flux, however, is so close to the detection limit that thestars cannot have an excess stronger than the photospheric flux densityat 60 μm. Of the stars younger than 400 Myr one in two has a disk;for the older stars this is true for only one in ten. We conclude thatmost stars arrive on the main sequence surrounded by a disk; this diskthen decays in about 400 Myr. Because (i) the dust particles disappearand must be replenished on a much shorter time scale and (ii) thecollision of planetesimals is a good source of new dust, we suggest thatthe rapid decay of the disks is caused by the destruction and escape ofplanetesimals. We suggest that the dissipation of the disk is related tothe heavy bombardment phase in our Solar System. Whether all starsarrive on the main sequence surrounded by a disk cannot be established:some very young stars do not have a disk. And not all stars destroytheir disk in a similar way: some stars as old as the Sun still havesignificant disks. Based on observations with ISO, an ESA project withinstruments funded by ESA Member States (especially the PI countries:France, Germany, The Netherlands and the UK) and with the participationof ISAS and NASA. Tables 2, 3 and 4 are also available in electronicform at the CDS via anonymous ftp cdsarc.u-strasbg.fr (130.79.128.5)} orvia http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/365/545 Detection of moving clusters by a method of cinematic pairs.Not Available Disappearance of stellar debris disks around main-sequence stars after 400 million years.Not Available Photometric Measurements of the Fields of More than 700 Nearby StarsIn preparation for optical/IR interferometric searches for substellarcompanions of nearby stars, we undertook to characterize the fields ofall nearby stars visible from the Northern Hemisphere to determinesuitable companions for interferometric phase referencing. Because theKeck Interferometer in particular will be able to phase-reference oncompanions within the isoplanatic patch (30") to about 17th magnitude atK, we took images at V, r, and i that were deep enough to determine iffield stars were present to this magnitude around nearby stars using aspot-coated CCD. We report on 733 fields containing 10,629 measurementsin up to three filters (Gunn i, r and Johnson V) of nearby stars down toabout 13th magnitude at V. A Second Catalog of Orbiting Astronomical Observatory 2 Filter Photometry: Ultraviolet Photometry of 614 StarsUltraviolet photometry from the Wisconsin Experiment Package on theOrbiting Astronomical Observatory 2 (OAO 2) is presented for 614 stars.Previously unpublished magnitudes from 12 filter bandpasses withwavelengths ranging from 1330 to 4250 Å have been placed on thewhite dwarf model atmosphere absolute flux scale. The fluxes wereconverted to magnitudes using V=0 for F(V)=3.46x10^-9 ergs cm^-2 s^-1Å^-1, or m_lambda=-2.5logF_lambda-21.15. This second catalogeffectively doubles the amount of OAO 2 photometry available in theliterature and includes many objects too bright to be observed withmodern space observatories. High resolution spectroscopy over lambda lambda 8500-8750 Å for GAIA. I. Mapping the MKK classification systemWe present an Echelle+CCD high resolution spectroscopic atlas (0.25Ä/pix dispersion, 0.43 Ä FWHM resolution and 20 000 resolvingpower) mapping the MKK classification system over the interval lambdalambda 8500-8750 Ä. The wavelength interval is remarkably free fromtelluric lines and it is centered on the near-IR triplet of Ca II, thehead of hydrogen Paschen series and several strong metallic lines. Thespectra of 131 stars of types between O4 and M8 and luminosity classes Ithrough V are included in the atlas. Special care was put in maintainingthe highest instrumental homogeneity over the whole set of data. Thecapability to derive accurate MKK spectral types from high resolutionobservations over the interval lambda lambda 8500-8750 Ä isdiscussed. The observations have been performed as part of an evaluationstudy of possible spectroscopic performances for the astrometric missionGAIA planned by ESA. Tables~3 and 4 are only available in electronicform at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5)or via http://cdsweb.u-strasbg.fr/ Abstract.html}\fnmsep\thanks{ Thespectra of the stars listed in Table~2 are also available in electronicform at the CDS or via the personal HomePagehttp://ulisse.pd.astro.it/Astro/Atlases/}\fnmsep\thanks{ Figures 3--28are only available in electronic form at http://www.edpsciences.com The ROSAT all-sky survey catalogue of the nearby starsWe present X-ray data for all entries of the Third Catalogue of NearbyStars \cite[(Gliese & Jahreiss 1991)]{gli91} that have been detectedas X-ray sources in the ROSAT all-sky survey. The catalogue contains1252 entries yielding an average detection rate of 32.9 percent. Inaddition to count rates, source detection parameters, hardness ratios,and X-ray fluxes we also list X-ray luminosities derived from Hipparcosparallaxes. Catalogue also available at CDS via anonymous ftp tocdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/Abstract.html Identification of lambda Bootis stars using IUE spectra. II. High resolution dataStars included in the catalogue of lambda Bootis stars by Paunzen et al.(\cite{Paunzen97}) with high resolution spectra (FWHM: 0.10-0.25 Ä)in the INES Archive of the International Ultraviolet Explorer satellite(IUE) are analyzed here in order to establish membership criteria forthe lambda Bootis group. Line-ratios of carbon to heavier elements (Si,Al, Ca) were adopted as criteria in the SWP range (1150-1980 Ä).For the LWP range (1850-3350 Ä), the intensity of metallic lines(Fe and Mg) was used. These criteria, together with those derived forlow resolution spectra, make the IUE Final Archive a powerful tool tofind new {lambda Bootis} candidates. A test of Hipparcos parallaxes on multiple stars.Not Available Averaged energy distributions in the stellar spectra.Not Available X-Ray Activity in the Open Cluster IC 4665We present the results of a joint ROSAT High Resolution Imager (HRI) andoptical investigation of the open cluster IC 4665. The ROSAT datacontains detections for 28 stellar sources in the field, including 22cluster members and candidate members spanning the color range -0.18<= (B-V)0 <= 1.63 (~B3-M3). Upper limits are given for theremaining members (or candidate members) in the HRI field. Keck HIRESspectra have been obtained that yield radial and rotational velocitymeasures, respectively, for faint, low-mass candidate members locatedwithin the field of the ROSAT HRI observation. In addition, photometryof possible optical counterparts to previously uncataloged X-ray sourcesin the HRI field is presented. The trends in X-ray properties with B-Vcolor in IC 4665 are found to be quite similar to that for other, morenearby young clusters such as the Pleiades and alpha Persei. Inparticular a maximum in normalized X-ray luminosity of log (LX/Lbol) ~=-3 is observed, beginning in the color range of (B-V)0 = 0.7-0.8. Thisis similar to the corresponding color range among Pleiades members, inagreement with the earlier estimate by Prosser & Giampapa that theage of IC 4665 is similar to the age of the Pleiades. The correlation ofrotation and X-ray emission levels is consistent with that in otheryoung clusters. Among the high-mass stars in IC 4665, five B stars aredetected as X-ray sources. Of these, one is a spectroscopic binary whilethe remaining objects are apparently single stars. The level ofintrinsic X-ray emission observed in the rapidly rotating (v sin i >200 km s-1), single B stars is consistent with an origin due to shockheating of the ambient medium by radiatively driven, rotationallyenhanced winds. On the basis of these observations and the results forother clusters, we argue that observed levels of X-ray emission inhigh-mass stars of log (LX/Lbol) > -6.0 are likely due to intrinsicemission associated with the B star itself rather than emission from alate-type secondary. Finally, our results suggest that the initialangular momentum distributions and the subsequent evolution of rotation,LX, and (LX/Lbol) occur in practically identical fashions in clusterswith ages <~100 Myr. Far-Infrared Photometry of Circumstellar Matter Around Intermediate Mass StarsNot Available The Sirius Supercluster and Missing Mass near the SunThe Hipparcos results confirm some 50 members of the Sirius superclusterin the Bright Star Catalogue. The resulting, well-definedcolor-luminosity array indicates an age of 4 x 10^8 yr from conventional(no overshoot) models. A comparison of the luminosity function obtainedfrom members in the Bright Star Catalogue, unbiased as to proper motion,and in the Catalogue of Nearby Stars (~27 pc), which are strongly biasedtoward large proper motions, reveals that either most of the smallproper-motion stars near the Sun remain to be identified or theluminosity function of the supercluster is drastically different fromthat of the field stars. A search for low-mass members in a 6 deg x 6deg field in the center of the Ursa Major cluster yielded 10 possiblemembers that lie on the white dwarf sequence, 5 mag below the mainsequence. Luyten (field LP 131) found 368 stars in this field withproper motion exceeding 0.08", and only 19 of these have a positionangle of their proper motion between 0 deg and 98 deg (i.e., thequadrant containing UMa cluster members). The assumption of clustermembership yields a parallax very close to that obtained for brightmembers. Aside from white dwarfs, the possibilities are discussed thatthese stars are (1) brown dwarfs, (2) members of a cluster in the lineof sight to UMa but 100 pc more distant, or (3) reflecting a chain ofvery unlikely coincidences. The ROSAT all-sky survey catalogue of optically bright main-sequence stars and subgiant starsWe present X-ray data for all main-sequence and subgiant stars ofspectral types A, F, G, and K and luminosity classes IV and V listed inthe Bright Star Catalogue that have been detected as X-ray sources inthe ROSAT all-sky survey; several stars without luminosity class arealso included. The catalogue contains 980 entries yielding an averagedetection rate of 32 percent. In addition to count rates, sourcedetection parameters, hardness ratios, and X-ray fluxes we also listX-ray luminosities derived from Hipparcos parallaxes. The catalogue isalso available in electronic form via anonymous ftp tocdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/Abstract.html

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The American administrative state has become a cost-benefit state, at least in the sense that prevailing executive orders require agencies to proceed only if the benefits justify the costs. Some people celebrate this development; others abhor it. For defenders of the cost-benefit state, the antonym of their ideal is, alternately, regulation based on dogmas, intuitions, pure expressivism, political preferences, or interest-group power. Seen most sympathetically, the focus on costs and benefits is a neo-Benthamite effort to attend to the real-world consequences of regulations, and it casts a pragmatic, skeptical light on modern objections to the administrative state, invoking public-choice theory and the supposedly self-serving decisions of unelected bureaucrats. The focus on costs and benefits is also a valuable effort to go beyond coarse arguments, from both the right and the left, that tend to ask this unhelpful question: “Which side are you on?” In the future, however, there will be much better ways, which we might consider neo-Millian, to identify those consequences: 1) by relying less on speculative ex ante projections and more on actual evaluations; 2) by focusing directly on welfare and not relying on imperfect proxies; and 3) by attending closely to distributional considerations–on who is helped and who is hurt.

From 1981 to the present, the American administrative state has become, to a significant extent, a cost-benefit state.1 Under prevailing executive orders, agencies must calculate the costs and benefits of proposed and final regulations, and to the extent permitted by law, may proceed only if the benefits justify the costs. These requirements have spurred, and helped make possible, life-saving regulations in a variety of domains, including clean air, motor vehicle safety, clean water, homeland security, public health, climate change, and occupational safety. At the same time, they have served as a check on, and an obstacle to, regulations that would cost a great deal and achieve very little.

Of course it is true that political considerations matter, even in a cost-benefit state. In Congress, cost-benefit analysis often takes a back seat, if it makes it into the room at all. In the executive branch, political convictions, dogmas, or perceived electoral considerations may trump the outcome of cost-benefit analysis, or make it an ex post justification or an afterthought, rather than a driver of decisions. Nonetheless, the analysis of costs and benefits, offered by technical specialists, often has a real impact on regulatory choices, pressing administrators in the direction of greater or less stringency, exposing new options, or offering a bright green GO! or a forbidding red STOP!

In terms of rigor, coverage, and accuracy, a great deal remains to be done. The fact that cost-benefit requirements do not apply to the “independent” agencies, such as the Federal Communications Commission, the Securities and Exchange Commission, and the Nuclear Regulatory Commission, is a continuing problem. Sometimes the numbers are based on guesswork, and there is continuing concern about whether before-the-fact estimates (of, for example, safety and health regulations) are reliable, or whether they are, on some occasions, a stab in the dark. Many people have argued for rigorous, ongoing evaluations, in which administrators test whether (for example) a regulation designed to increase food safety, or to protect against occupational injuries, is actually having its intended effect, and whether it is doing better or worse than expected. They are right to make that argument.

Despite the continuing challenges, the emergence of the cost-benefit state is a remarkable achievement. It means that the role of dogmas, intuitions, and interest groups has diminished and that within the executive branch, at least, regulators have often focused insistently on the human consequences of what they are proposing to do. To a significant extent, the cost-benefit state has been a check on “expressivism,” in which public officials, on either the left or the right, act to express abstract values without exploring whether particular initiatives would actually have good or bad consequences. To the extent that the consequences of regulations are genuinely good (because, say, they prevent hundreds or thousands of deaths), the rise of the cost-benefit state casts a new light on some prominent and high-minded critiques of modern administration–for example, that it is a product of unelected bureaucrats, a tribute to the power of well-organized private groups, a reflection of monied interests, an unacceptable abdication of legislative authority, or a product of government's efforts to expand its own power.

To be sure, each of these critiques must be met on its own terms. But if (for example) a motor vehicle safety regulation from the Department of Transportation, authorized by Congress, is preventing three hundred deaths annually and costing just $40 million, it would not seem that there is good reason for complaint, and the same is true if the Environmental Protection Agency (EPA) is finding ways to reduce greenhouse gases significantly and at modest cost. Indeed, many regulations, under both Republican and Democratic administrations, have delivered massive net benefits (understood as benefits minus costs). It is not unusual to find that in a given year, the monetized benefits of regulations (including the benefits in terms of preventing illnesses, accidents, and premature deaths) exceed the monetized costs by many billions of dollars. (The Trump administration was an outlier; because it issued so few regulations, the annual costs of what it did were very low, and so were the annual benefits.) Under favorable conditions, the use of cost-benefit analysis can provide safeguards against decisions based on feelings, hopes, presumptions, perceived political pressures, appealing but evidence-free compromises, broad aspirations, guesses, or the wishes of the strongest people in the room. But the administrative state should do better still. It needs to focus directly on human welfare. It should see cost-benefit analysis as a mere proxy for welfare, and an imperfect one to boot. It needs to investigate welfare itself, and to explore what that idea is best understood to mean. It needs as well to focus on distributional considerations–on who is helped and who is hurt. To see the underlying problems, consider a realistic if highly stylized example. Suppose that the Environmental Protection Agency is considering a new regulation designed to reduce levels of particulate matter in the ambient air. Suppose that the total annual cost of the regulation would be$900 million. Suppose that the monetized mortality benefits would be higher than that–because, say, the regulation would prevent one hundred deaths, each valued at $10 million. (This is a hypothetical number; as of 2021, prominent federal agencies valued a statistical life at about$11 million.) Suppose as well that if the EPA includes morbidity benefits (in the form of nonfatal illnesses averted), the regulation would produce an additional $350 million in benefits, meaning that the monetized benefits ($1.35 billion) are significantly higher than the monetized costs ($900 million). At first glance, the cost-benefit analysis suggests that the regulation is an excellent idea, and that the EPA should go forward with it. Now assume four additional facts. First, the mortality benefits of the regulation would be enjoyed mostly by older people: those over the age of eighty. Second, the rule would have significant disemployment effects, imposing a statistical risk of job loss on a large number of people, and ultimately causing three thousand people to lose their jobs. Third, the EPA believes that the overwhelming majority of those three thousand people would find other jobs, and probably do so relatively soon, but it does not have a great deal of data on that question and it cannot rule out the possibility of long-term job loss for many people. Fourth, the mortality and morbidity benefits would be enjoyed disproportionately by low-income communities and by people of color. In accordance with standard practice, the EPA does not include any of those further facts in its cost-benefit analysis. If the goal is to promote social welfare, it would be far too simple for the EPA to conclude that, because the monetized benefits exceed the monetized costs, it should proceed with the regulation. One question is whether and how to take into account, in welfare terms, the relatively few additional life-years that the regulation will generate. In those terms, is a rule that “saves” people over eighty to be deemed equivalent to one that “saves” an equivalent number of people who are (say) under thirty? And what are the welfare consequences of the$900 million expenditure? Suppose that, concretely, the admittedly high cost will be spread across at least two hundred million people, who will be spending, on average, a little over $4 annually for the regulation. What are the welfare consequences of that modest expenditure? Might they be relatively small? (The answer is emphatically yes. Most people will lose essentially no welfare from an annual$4 loss.)

A further question is the disemployment effect. We know that in terms of subjective welfare, it is extremely bad to lose one's job.2 People who lose their jobs suffer a lot: Job loss can severely harm one's self-worth and experience of daily life. A sudden loss of income can threaten housing and food security, often causing disruptions to family life and schooling. A loss of a job also creates a nontrivial long-term loss in income.3 If you are out of work for a year, the economic toll might be very high over a lifetime. We know that a long-term loss of employment has more severe adverse consequences than a short-term loss, but both are bad. Shouldn't those welfare effects be included?

Yet another question is the distributional impact. If the health benefits of regulation would be enjoyed mostly by members of low-income groups, and particularly by people of color, might that matter? We might think that even if the rule does not have significant net welfare benefits, or even if it has some net welfare costs, it is nonetheless desirable, if and because it increases equality. The interest in environmental justice focuses on the very real possibility that wealthy people might be the disproportionate beneficiaries of polluting activity and that poor people might bear most of the costs. (In the context of air pollution, that appears to be true.)

These considerations suggest that while monetized costs and benefits tell us a great deal, they do not tell us everything that we need to know. On welfare grounds, a rule might not make sense even if the monetized benefits are higher than the monetized costs, and a rule might make sense even if the monetized costs are higher than the monetized benefits. In addition, we should want to consider distributional effects. To be sure, a rule that costs $1 billion and that provides benefits of$100 would not be a good idea even if the wealthy pay that $1 billion and poor people receive that$100. But if a rule costs $1 billion and delivers$950 million in benefits, we might want to go forward with it if the cost is diffused among a large number of wealthy people, and if the benefit is enjoyed by (for example) coal miners whose lives are at stake.

Now suppose that the Department of Transportation is considering a regulation that would require all new automobiles to come equipped with cameras, so as to improve rear visibility and thus reduce the risk of backover crashes.4 Suppose that the total estimated annual cost of the regulation is $1.2 billion (reflecting an average added cost of$300 per vehicle sold over the relevant time period). Suppose that the regulation is expected to prevent sixty deaths annually, for monetized annual savings of $540 million, as well as a number of nonfatal injuries and cases of property damage, for additional annual savings of$200 million. On the basis of these numbers, the Department is inclined to believe that the benefits of the rules are significantly lower than the costs.

At the same time, suppose that the Department is aware of four facts that it deems relevant, but that it is not at all sure how to handle. First, a majority of the deaths that the regulation would prevent would involve young children, between the ages of one and five. Second, a majority of those deaths would occur as a result of the driving errors of their own parents, who would therefore suffer unspeakable anguish. Third, the cost of the rule would be diffused across a large population of new car purchasers, who would not much notice the per-vehicle cost. Fourth, the cameras would improve people's driving experience by making it much easier for them to navigate the roads, even when it does not prevent crashes. (The Department speculates that many consumers do not sufficiently appreciate this improvement when deciding which cars to buy.) Is it so clear, in light of these four facts, that the agency should not proceed? That is not a hard question. The answer is: no. That answer suggests the importance of considering variables that are difficult or perhaps impossible to quantify. (How exactly to do that is a hard question.)

In principle, cost-benefit analysis is best defended as the most administrable way of capturing the welfare effects of policies (including regulations). But if we actually knew those effects, in terms of people's actual welfare (suitably specified), and thus could specify the actual consequences of policies for welfare (again, suitably specified), we would not have to trouble ourselves with cost-benefit analysis. An initial problem is that cost-benefit analysis depends on willingness to pay, and people might be willing to pay for goods that do not have substantial positive effects on their welfare (and might be unwilling to pay for goods that would have substantial positive effects). Willingness to pay is based on a prediction, and at least some of the time, people make mistakes in forecasting how various outcomes will affect their lives (and make them feel). Call them welfare forecasting errors. You might think that if you do not get a particular job, or if your favorite sport team loses a crucial game, or even if someone you really like refuses to date you, you will be miserable for a good long time. But chances are that you are wrong; you will recover much faster than you think. The basic point applies to the administrative state and its choices. People might make welfare forecasts with respect to calorie consumption or exposure to certain risks, and those forecasts might go wrong. If administrators rely on welfare forecasts as reflected in willingness to pay, they might incorporate and hence propagate errors.

A separate problem involves the incidence of costs and benefits, which can complicate the analysis of welfare effects, even if we put “pure” distributional considerations to one side. Suppose that a regulation would impose $400 million in costs on relatively wealthy people and confer$300 million in benefits on relatively poor people. Even if the losers lose more than the gainers gain in monetary terms, we cannot exclude the possibility that the losers will lose less than the gainers gain in welfare terms.

An additional problem is that because willingness to pay depends on ability to pay, it can be a poor measure of welfare effects. A very rich person might be willing to pay a lot (say, $2,000) for a good from which she would not get a lot of welfare. (After all, losing$2,000 is a trivial matter, if you are very rich.) A very poor person might be willing to pay only a little (say, $20 and no more) for a good from which she would get a lot of welfare. (After all, losing$20 is no trivial matter, if you are very poor.) These points do not mean that a very rich person should be prevented from paying that large amount for that good, or that a very poor person should be forced to pay more than that small amount for that good. (People who like regulation often miss the latter point in particular.) But they emphatically do mean that if a very poor person, or simply a poor person, is willing to pay only a small amount to avoid a mortality risk, or to get some benefit (say, an unlawfully present citizen seeking “deferred action” from the U.S. government), that small amount is not a good measure of the welfare effects.

The most general problem is that whenever agencies specify costs and benefits, the resulting figures will inevitably have an ambiguous relationship with what they should care about, which is welfare. To be sure, it is possible that some of the problems in the two cases I have given could be significantly reduced with improved cost-benefit analysis. If children should be valued differently from adults, and elderly people differently from younger, cost-benefit analysis might be able to explain why and how. Perhaps parental anguish could be monetized as well. (Why, you might ask? It is a fair question. The answer is to figure out how to weigh both sides of the ledger; without that, how can a regulator make a sensible decision?) The same might well be true, and might more readily be true, of the increased ease of driving. But even the best proxies remain proxies, and what matters most is welfare itself.

In recent years, social scientists have become greatly interested in measuring welfare. One of their techniques is to study “self-reported well-being,” meaning people's answers to survey questions about how satisfied they are with their lives. The promise of this technique is that it might be able to offer a more direct, and more accurate, measure of welfare than could possibly come from an account of costs and benefits (especially if that account depends on willingness to pay).5 Suppose that we agree with economist Paul Dolan that welfare largely consists of two things: 1) people's feelings of pleasure (broadly conceived) and 2) people's feelings of purpose (also broadly conceived).6 People might enjoy watching sports on television, but they might not gain much of a sense of purpose from that activity. Working for a good cause (consider working for a nonprofit or for a government whose leaders you admire) might not be a lot of fun, but it might produce a strong sense of purpose.

If pleasure and purpose matter, and if we want to measure them, we might be able to ask people about those two variables. How much pleasure do people get from certain activities? How much of a sense of purpose? Dolan has in fact asked such questions, with illuminating results.7 We are learning a great deal about what kinds of activities are pleasurable or not, and also about what kinds of activities seem to give people a sense of purpose or meaning. In the abstract, what we learn seems to tell us a lot about people's welfare, and it might offer a more direct and accurate account than what emerges from an analysis of costs and benefits. The reason is that measures of pleasure and purpose offer information about people's actual experience of their lives, rather than a projection as measured by money, and the former seems to be what most matters.

With respect to subjective well-being, the most popular existing measures take two forms. First, researchers try to assess people's “evaluative” welfare by asking questions about overall life satisfaction (or related concepts, such as happiness).8 With such measures, it is possible to test the positive or negative effects of a number of life events such as marriage, divorce, disability, and unemployment.9 Second, researchers try to assess people's “experienced” welfare, through measures of people's assessments of particular activities (working, commuting, being with friends, watching television).10

In fact, researchers have uncovered some systematic differences between people's overall evaluations and their assessments of their particular experiences.11 Marital status is more closely correlated with experienced well-being than with evaluative well-being, though there is conflicting evidence on this point.12 French people report significantly lower levels of satisfaction with their lives than Americans, but the French appear to show equal or even higher levels of experienced well-being.13 (Psychologist Daniel Kahneman has suggested a partial explanation: in France, if you say you are happy, you are superficial; in the United States, if you say you are unhappy, you are pathetic.) Health states are more closely correlated with experienced well-being, though they also affect evaluative well-being.

How can the choice be made between the two measures?14 The emerging consensus is that useful but different information is provided by each. On one view, questions about experienced welfare focus people on their existing emotional states, and thus provide valuable information about those states. By contrast, questions about evaluative welfare encourage people to think about their overall goals or aspirations. On this view, evaluative welfare “is more likely to reflect people's longer-term outlook about their lives as a whole.”15 If this is so, then the two measures do capture different kinds of values, and both are important. But it is not clear that the emerging consensus is correct, for a critical question remains: do people's answers to questions about evaluative well-being in fact reflect their broader aspirations, or do they represent an effort to summarize experienced well-being (in which case the latter is the more accurate measure)?

True, the idea of “welfare” leaves a great deal of ambiguity, and if it is invoked for policy purposes or by governments, any particular account is highly likely to end up in contested terrain.16 As made clear by Dolan (not to mention Aristotle, John Stuart Mill, and Amartya Sen), a neo-Benthamite measure, purely hedonic and focused only on pleasure and pain, would be inadequate; people's lives should be meaningful as well as pleasant. But even if we adopt a measure that goes beyond pleasure to measure a sense of purpose as well, we might be capturing too little. We might be ignoring qualitative differences among goods and the general problem of incommensurability.

We value some things purely or principally for use; consider hammers, forks, or money. We value other things at least in part for their own sake; consider knowledge or friendship. But that distinction captures only part of the picture. Intrinsically valued things produce a range of diverse responses. Some bring about wonder and awe; consider a mountain or a work of art. Toward some people, we feel respect; toward others, affection; toward others, love. (There are of course qualitative differences among different kinds of love.) Some events produce gratitude; others produce joy; others are thrilling; others produce a sense of wonder; others make us feel content; others bring about delight. Some things are valued if they meet certain standards, like a musical or athletic performance, or perhaps a pun. In this regard, Mill's objections to Bentham are worth quoting at length:

Nor is it only the moral part of man's nature, in the strict sense of the term–the desire of perfection, or the feeling of an approving or of an accusing conscience–that he overlooks; he but faintly recognizes, as a fact in human nature, the pursuit of any other ideal end for its own sake. The sense of honour, and personal dignity–that feeling of personal exaltation and degradation which acts independently of other people's opinion, or even in defiance of it; the love of beauty, the passion of the artist; the love of order, of congruity, of consistency in all things, and conformity to their end; the love of power, not in the limited form of power over other human beings, but abstract power, the power of making our volitions effectual; the love of action, the thirst for movement and activity, a principle scarcely of less influence in human life than its opposite, the love of ease. … Man, that most complex being, is a very simple one in his eyes.17

These points suggest the importance of having a capacious conception of welfare, one that is alert to the diverse array of goods that matter to people. Consistent with Mill's plea, a large survey by the economist Daniel Benjamin and coauthors tests people's concern for a list of factors that includes not only “measures widely used by economists (e.g., happiness and life satisfaction),” but also “other items, such as goals and achievements, freedoms, engagement, morality, self-expression, relationships, and the well-being of others.”18

The central and important (though not especially surprising) result, compatible with Mill's point, is that people do indeed care about those other items.19 The perhaps ironic conclusion is that, if measures of reported well-being neglect those items, they will end up losing important information that cost-benefit measures ought to be able to capture. A significant advantage of the willingness-to-pay measure is that it should, in principle, take account of everything that people care about, including those things that matter for Mill's reasons. If people value cell phones because they want to connect with their children, or if they want to save (rather than spend) money so they can give it to impoverished children, or if they want to spend money on a vacation because of their love of nature, their concerns, however diverse in qualitative terms, should be adequately captured by the willingness-to-pay criterion, however unitary.

That is a point for cost-benefit analysis. Notwithstanding its apparent crudeness, and notwithstanding the simplicity of the monetary measure, it honors qualitatively diverse goods that people care about for diverse reasons. In that way, it is not simple at all, and for that reason, cost-benefit analysis has advantages over some measures of happiness or subjective welfare. Nonetheless, that form of analysis cannot have priority over excellent or full measures of welfare. What is required are measures that are sufficiently reflective of the diverse set of goods that matter to people but that avoid the various problems, signaled above, of cost-benefit analysis.

With respect to regulatory policy, the largest problem with invoking self-reported well-being is this: even if such surveys provide a great deal of information, we cannot easily “map” any particular set of regulatory consequences onto changes in welfare.

Although we are learning a great deal about what increases and what decreases welfare, what we are learning is relatively coarse; it frequently involves the consequences of large life events, such as marriage, divorce, and unemployment.20 We do not know nearly enough about how to answer hard questions about the welfare effects of health, safety, and other regulations. For example: 1) How much happier are people when the level of ozone in the ambient air is decreased from seventy parts per billion to sixty parts per billion? 2) For the median person, what is the welfare effect of having to spend $50 or$100 or \$300 on a particular regulatory initiative, noting that the money could have been used for other purposes? 3) What are the welfare effects of giving unlawful noncitizens in the United States deferred action, meaning that they will not be deported and will be authorized to work? 4) In terms of “welfare units,” how should we think about a loss of a job, or a life-year? Should we use those units or some other kind of unit (monetary?) in conducting analyses on the basis of studies of self-reported well-being? If we use welfare units, what, exactly, is the relevant scale?

Return to the two problems with which I began. We have seen that in terms of welfare, cost-benefit analysis, at least in its current form, may not adequately handle: 1) unusually large or unusually small numbers of life-years saved; 2) adverse unemployment effects; 3) questions about the welfare effects of small economic losses faced by large populations; 4) intense emotions associated with certain outcomes, such as parental anguish (or fear); and 5) hedonic benefits associated with increased ease and convenience. We have also seen that cost-benefit analysis does not capture distributional impacts, and that they might greatly matter. As I have suggested, improved forms of cost-benefit analysis might be able to reduce these problems (and cost-benefit analysis can of course be complemented with other inquiries; we might engage in that form of analysis and deal with distributional impacts separately). But ideally, we would want to know about welfare itself. The problem is that measures of self-reported well-being are far too crude to enable us to do that.

No one should doubt that cost-benefit analysis itself presents serious challenges, sometimes described under the rubric of “the knowledge problem”: agencies have to compile a great deal of information to make sensible extrapolations. But to map regulatory outcomes onto self-reported well-being, the challenges are far more severe. Does this conclusion mean that today and in the near future, regulators should rest content with cost-benefit analysis, and put entirely to one side, as speculative and unreliable, whatever we might learn from directly considering welfare? That would be too strong. Most important, disemployment effects deserve serious consideration, not least because of the significant adverse welfare effects of losing one's job. It is also relevant to know whether a regulation would protect children, and hence provide a large number of life-years, or instead (and this is a far more controversial question) protect older people, and hence provide a relatively smaller number of life-years. The Department of Transportation was correct to emphasize that its rear visibility rule would disproportionately protect children.

It is also possible that a large cost, spread over a very large population, might turn out to have relatively modest adverse effects on welfare. Agencies should consider this possibility, especially in cases in which costs and benefits are otherwise fairly close. And if agencies would (for example) help people who suffer from mental illness of one or another kind, the welfare gain might be substantial, even if the benefits cannot be adequately captured in willingness-to-pay figures. Distributional effects should also be considered; they matter.

Emphasizing the promise of research on subjective well-being, economist Raj Chetty contends: “Further work is needed to determine whether and how subjective well-being metrics can be used to reliably measure experienced utility, but they appear to offer at least some qualitative information on ex post preferences [that] can help mitigate concerns about paternalism in behavioral welfare economics.”21

Chetty's conclusion is sound, but it could be much stronger. Work on subjective well-being can serve not only to mitigate concerns about paternalism but, at least on occasion, to inform analysis of the welfare effects of regulations (and policies in general). At present, inquiries into subjective well-being are too coarse to provide a great deal of help to administrators, and cost-benefit analysis is the best proxy they have for (much of) what matters. But it cannot possibly tell us everything that we need to know. In the fullness of time, it will be supplemented or perhaps even superseded by a more direct focus on welfare.

1

See Cass R. Sunstein, The Cost-Benefit Revolution (Cambridge, Mass.: The MIT Press, 2018).

2

See Jonathan Masur and Eric A. Posner, “Regulation, Unemployment, and Cost-Benefit Analysis,” Virginia Law Review 98 (3) (2012): 580, who refer to a number of sources, including Andrew E. Clark and Andrew J. Oswald, “Unhappiness and Unemployment,” The Economic Journal 104 (424) (1994): 648, 650–651, finding that unemployment is associated with significantly lower self-reported mental well-being; William T. Gallo, Elizabeth H. Bradley, Joel A. Dubin, et al., “The Persistence of Depressive Symptoms in Older Workers Who Experience Involuntary Job Loss: Results from the Health and Retirement Survey,” The Journal of Gerontology, Series B: Psychological Sciences and Social Sciences 61 (4) (2006): S221, finding that older, lower net-worth workers who lose their jobs are more likely to suffer from depression than those who do not; and Knut Gerlach and Gesine Stephan, “A Paper on Unhappiness and Unemployment in Germany,” Economics Letters 52 (3) (1996): 325, finding that unemployment reduces life satisfaction beyond what would be expected from the loss of income.

3

See Masur and Posner, “Regulation, Unemployment, and Cost-Benefit Analysis,” 580.

4

For detailed discussion, see Cass R. Sunstein, “Rear Visibility and Some Unresolved Problems for Economic Analysis,” Journal of Benefit-Cost Analysis 10 (3) (2019): 317. The example is real but the numbers are definitely not. For the actual numbers, see “Federal Motor Vehicle Safety Standards; Rear Visibility: A Rule by the National Highway Traffic Safety Administration,“ Federal Register 79 (66) (2014): 19178–19250.

5

Overviews can be found in Paul Dolan, Happiness by Design: Change What You Do, Not How You Think (New York: Avery, 2014); Daniel Kahneman, Edward Diener, and Norbert Schwarz, eds., Well-Being: Foundations of Hedonic Psychology (New York: Russell Sage Foundation, 2002); and Daniel Gilbert, Stumbling on Happiness (New York: Knopf, 2006). I am bracketing the question whether it is best to have a subjective or objective account of welfare; certainly subjective welfare matters, even if we adopt an objective account. Valuable discussion can be found in Matthew Adler, Well-Being and Fair Distribution: Beyond Cost-Benefit Analysis (Oxford: Oxford University Press, 2011).

6

See Dolan, Happiness by Design.

7

See ibid.

8

See Richard Layard, Happiness: Lessons from New Science (New York: Penguin Books, 2006).

9

For discussion, see Cass R. Sunstein, “Illusory Losses,” The Journal of Legal Studies 37 (S2) (2008): S157.

10

Arthur A. Stone and Christopher Mackie, eds., Subjective Well-Being: Measuring Happiness, Suffering, and Other Dimensions of Experience (Washington, D.C.: National Academies Press, 2013).

11

Daniel Kahneman and Jason Riis, “Living, and Thinking about It: Two Perspectives on Life,” in The Science of Well-Being, ed. Felicia A. Huppert, Nick Baylis, and Barry Keverne (Oxford: Oxford University Press, 2005).

12

Stone and Mackie, Subjective Well-Being, 33.

13

See Kahneman and Riis, “Living, and Thinking about It.”

14

See ibid. Dolan, Happiness by Design, argues for the priority of experienced well-being. Stone and Mackie conclude that multiple dimensions exist and are worth measuring, see Subjective Well-Being, 32.

15

Stone and Mackie, Subjective Well-Being, 33. See also the discussion of “eudaimonic well-being,” drawn from ideas about human flourishing, ibid., 18.

16

Adler, Well-Being and Fair Distribution.

17

See John Stuart Mill, “Bentham,” in Utilitarianism and Other Essays, ed. Alan Ryan (New York: Penguin Books, 1987), 132.

18

See Daniel Benjamin, Ori Heffetz, Miles S. Kimball, and Nichole Szembrot, “Beyond Happiness and Satisfaction: Toward Well-Being Indices Based on Stated Preference,” American Economic Review 104 (9) (2014): 2698.

19

Ibid.

20

For a valuable discussion, see W. Kip Viscusi, “The Benefits of Mortality Risk Reduction: Happiness Surveys vs. the Value of a Statistical Life,” Duke Law Journal 62 (8) (2013): 1735.

21

See Raj Chetty, “Behavioral Economics and Public Policy: A Pragmatic Perspective,” American Economic Review 105 (5) (2015): 25.

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15 terms. Function Notation, Mathispower4u. function -- let me say h of x -- h of x could be defined as -- it literally The function f of x is graphed. 0 is not a -- x equals to 0 is not a member of that If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The function never goes below 0. This function definition does not tell us what to actually do with 0. If x satisfies this So in this case if -- let me see -- that's my function f. If I were to input the number 3. inequality right over here. So we write down these, these big ideas. ... Logarithmic and Exponential Functions and Equations. Check all that apply. This algebra 2 and precalculus video tutorial focuses on graphing exponential functions with e and using transformations. It gives a question mark. So here we are in putting a y it to function g and we're gonna output g of y. If $$\log_{b}(x)=y$$ then $$y$$ can be any real number. and I can even pick the values in So the domain, the domain here, the domain of h is literally -- it's just literally The function f of x is graphed. The graph of f is the graph of the equation y = f(x). This is the domain -- the domain of a function -- Actually let me write that out. Solving Rational Equations, Mathispower4u. Well, we go up here. It does equal 0 right over f of pi -- when x is pi, we're going to output Or we could say negative 6 Let's do a few more. We just think this is kind of the the So this right over So this hopefully starts to give you a flavor of why we care about to the domain. the exception. could be defined as, well h of x is gonna be 1 if x is equal to pi and it's equal to 0 if, if, x is equal to 3. traditional principal root operator. thinking about the x's for which this So you could say the domain here, we For any small positive number y, we can make bx smaller than yjust by here looks like it's 0. we have the function -- let's say we have the function f of x is equal to 2 over x. If our input was pi, then we input into our function and then We could say, let's say we which the function is defined. Negative 2 is less than or between negative 6, including negative So negative 2 is less than or It never gets above 8, but it here, negative 1 is less than or equal to x The domain of a function A domain of a function is the set of all inputs -- inputs over which the function is defined -- over which the function is defined, If you're seeing this message, it means we're having trouble loading external resources on our website. 6, and positive 7, including positive Determining the range of a function (Algebra 2 level). equal to 2 over 3. between these integers. So hopefully where the function is defined. Review: Laws of Exponents, domain and range of one-to-one functions and their inverses. At negative 1, it If you're behind a ... Domain and Range of a Function. We need something that -- On Tuesday, you will do a review of exponential functions. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. could say the domain here is the set of all y's that are members of Linear & Exponential Functions. Well, it hasn't defined. What is the set of all inputs over which this function g Learning Objectives: Study and understand the basic exponential function $$f(x) = b^x, b > 0, b \neq 1$$ Domain, Range, Intercepts, Asymptote and Graph So the domain of this Worked example: domain and range from graph. What is its range? So the way it's graphed First we study exponential functions. If you give me an x anywhere Sal covers many different kinds of functions and shows how to determine their domain. equals negative 9 or x equals negative 8 and 1/2 or When x equals 7, f I said OK Subsection 4.1.5 The Graph of a Logarithmic Function. here? Domain of advanced functions (video) | Khan Academy. And it's defined all the So on and so forth, If you're seeing this message, it means we're having trouble loading external resources on our website. The graph of f is the graph of the equation y = f(x). Khan Academy is a 501(c)(3) nonprofit organization. f of x is defined for x all thing, or only whole numbers, or natural numbers, or positive numbers, and negative the function is at that point. All of the site's resources are available to anyone. this is starting to make some sense -- You're all used to a function that is ... Domain and Range of a Function ... Graphing Radical Functions (cube root) Exponential Functions. The function f(x) = 0.11(3)x is reflected over the x-axis to produce function g(x). Review: Laws of Exponents, domain and range of one-to-one functions and their inverses. (A2.7I) I can analyze data to select the appropriate model from among linear, quadratic, and exponential models. Helpful Videos: Khan Academy – Sequences; Helpful Explanations: Purple Math – Sequences; Module 2: Linear and Exponential Functions. always have to use f's and x's. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. just have to move up above that number, So, for example, let's say that The Range of a Function is the set of all y values or outputs i.e., the set of all f(x) when it is defined.. We suggest you read this article “9 Ways to Find the Domain of a Function Algebraically” first. Domain is the set of all inputs over Graphing Exponential Functions; Solving Exponential Equations (exponents properties) Solving Exponential Equations (using logarithms) Logarithmic Functions This is kind of fun. the function tells us what we need to output. It's not defined for x Khan Academy is a 501(c)(3) nonprofit organization. Just be clear, we don't In Module 2, students will work with continuous linear and exponential functions in various representations (context, table, equation, graph). equal to negative 6. function f would be all real numbers except for x equals 0. have g of y is equal to the square root of y minus 6. negative, our traditional principal root operator here is not defined. is defined? Now let's make this a little bit So we'll see that as we do So the domain for this f First we study exponential functions. We're able to find the output And if xis negative then bx = 1 b x which is the reciprocal of a positive number and so is still positive. You can take any x value If we input 0 then But I want to do something interesting. The domain of y is $\left(-\infty ,\infty \right)$. And then the highest y The function f of x is graphed. Exponential and logarithmic functions (Precalculus) 5 Determine the function that give the number of bacteria after t hour and find the Get more practice. This will help you to understand the concepts of finding the Range of a Function better.. You could see a could write this as 2 over pi. ...Next lesson. is defined for negative 2. if this was a negative number, how would you take the principal defined for two input. You could even see functions that are divided fairly exotic ways. the set of y values. x equals negative 8. cannot be 0 because we don't know -- this definition is undefined when or the function has defined outputs over which the function has defined outputs. Recommended to you based on your activity and what's popular • Feedback Let's do a few more. we know that if you input 3 into it h of 3, when x equals 3, you're going to pretty easily. 7, and you just have to see-- you These are the only valid inputs. Both functions are one-to-one and are inverses of each other. Sort by: ...That doesn't mean that the math of the equations wouldn't give you a number for t = 0.50 hours, but … If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. It's not defined for f of negative 1 is negative 5. or equal to 7. Our mission is to provide a free, world-class education to anyone, anywhere. The function f of x is graphed. definition would say f of 0 be 2 over 0, but 2 over 0 is defined for any x that is greater than or scarfox166. Our discoveries about the domain and range of a logarithmic function lead us to wonder what the graph might look like. I found an answer from www.khanacademy.com. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. , it could be the set of -- I gonna put curly brackets like that. The Khan Academy is an organization on a mission. Some are defined domain -- such that x does not -- does not equal 0. wherever you are, to find out what the value of y that are greater than or equal to 6. going to be the the two valid inputs that x can be are 3 and pi. The graph of f is the graph of the equation y = f(x). Both functions are one-to-one and are inverses of each other. Well, f of x is condition right over here, the function is defined. We don't see it's graphed here. function definition is 8. f of 7 is 8. any of these values. log10A = B In the above logarithmic function, 10is called asBase A is called as Argument B is called as Answer here and it takes inputs, and for a given input, it's going to Well, exact similar argument. What is its domain? So once again, this function 5 Steps to Find the Range of a Function, greater than or equal, such that they're also greater than or equal to 6. It's going to be Therefore y= bx is always positive. Learning Objectives: Study and understand the basic exponential function $$f(x) = b^x, b > 0, b \neq 1$$ Domain, Range, Intercepts, Asymptote and Graph So let's check our answer. know how to figure that out. is less than or equal to 8. This video teaches about domain and the range of a function. member So this little symbol means a This mission consists of many topics that build off of the Algebra I Math Mission such as systems of equations, inequalities, functions, exponential and logarithmic functions, logarithms, and many more topics. Usually a logarithm consists of three parts. Let's have a little bit of a review of what and we write real numbers -- we write it with this kind of double stroke right over here. If this becomes a Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. member of the real numbers. is less than or equal to x, which is less than Function g(x) is then reflected over the y-axis to produce function h(x). Well, f of 3 that we're going to output -- we have, we Donate or volunteer today! Worked example: domain and range from graph. defined this way. Introduction to the domain and range of a function. So this gets to the essence of what domain is. defined for any x that satisfies this double In Module 2, students will work with continuous linear and exponential functions in various representations (context, table, equation, graph). https://www.khanacademy.org/.../v/domain-and-range-from-graphs starts getting defined. is equal to negative 9? in particular -- so the domain for this one -- if I want (Exponential growth describes the current pandemic of COVID-19 sadly ) Wednesday – Friday, you will review functions and function notation. value or the highest value that f of x obtains in this numbers. Before working with graphs, we will take a look at the domain (the set of input values) for which the logarithmic function is defined. Improve your math knowledge with free questions in "Domain and range of exponential functions: graphs" and thousands of other math skills. Exponential and logarithmic functions Evaluate Logarithms Practice Khan Academy » Exponential And Logarithmic Functions … What is its domain? And then it keeps So this function is not defined here. of x is equal to 5. if you input pi into it we know you're gonna output 1, and It's not all functions are defined over all real numbers. This is undefined. the way from x equals negative 6 all the The Algebra II Math Mission is a set of skills on Khan Academy. f of x is equal to 5. What is its domain? So f of x-- so 0 is less equal to x, which is less than or equal to 5. more concrete by do some more examples So more examples we do, hopefully the clearer this will become. is the entire function definition for f of x. Well, x can be a y values fall into? you put the input as 0 So x is a member of the real numbers, here. 1 is less than or equal to x. We're a not-for-profit with the goal of changing education for the better by providing a free world-class education to anyone anywhere. Rewrite this -- 2 over 0. Math: Unit Test. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. is less than or equal to 7, the function is ... Identify the domain and range of a function. So 0 is less than f of x, which If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. It hasn't defined. Introduction to the domain and range of a function. f of negative 4 is 0. This will address standards 8.F.A.1 and HS.IF.B.6: I can understand that a function is a rule that assigns to each input exactly one output. If you're seeing this message, it means we're having trouble loading external resources on our website. So far we know: The domain is limited to only positive input values. Where do all of the In fact, the range is all real numbers. It's going to be equal to 2 over 3. We've defined it right over here. It could be most of the real numbers except it than or equal to f of x. A simple exponential function like f (x) = 2 x has as its domain the whole real line. as long as whatever we have under the radical right over here is non-negative. If you, we know h of -- we know h of pi -- able to find an output. So what is the domain here? F.IF 5 1.2 Translations F.IF.7A 1.3 Writing Equations F.BF.3 1.4 Reflections F.BF 3 1.5 Domain Restriction Graphing F.IF 5 1.6 Evaluating & Graphing Piecewise-defined Functions These are the only two numbers over which this function is actually defined. So the domain of this definition tell us what we need to output? It only starts getting defined be the set of these curly brackets. undefined. way up to x equals 7, including x equals 7. for x is negative 9, negative 8, all the way down or all the way So for example, if So we're able, for that input, we're way to x equals 7. up I should say to negative 1. f of pi, which is equal to 2 over pi. Our mission is to provide a free, world-class education to anyone, anywhere. y is to be greater than or equal 6. Now what's the domain Graphing Linear functions using intercepts from Khan Academy ONLY ALGEBRA 2 Graphing a Quadratic Function from Mash Up Math Graphing a Quadratic Function from YouTube Key Features of a Quadratic Function from YouTube MATH ANALYSIS Review of Logarithms retrieved from Khan Academy Introduction to Logarithms - Video 1 retrieved from Khan Academy If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. So let's say we have another function. 1 Mission specifics 1.1 Mission Foundations 1.2 Manipulating functions 1.3 Introduction to complex … Domain and Range of Exponential and Logarithmic Functions Recall that the domain of a function is the set of input or x -values for which the function is defined, while the range is the set of all the output or y -values that the function takes. And I encourage you to pause the video and think about it. f of negative 2 is negative 4. f of negative 1 is negative 3. Well it's going to be defined Well, let's see. ... Reflections of Exponential Functions. In this article, you will learn. Graph exponential functions and find the appropriate graph given the function. But it can't be any real number. Determine the domain and range of the function f of x is equal to 3x squared plus 6x minus 2. If you're seeing this message, it means we're having trouble loading external resources on our website. for only a small subset of real numbers, or for some other So we And, I can take any real number, square it, multiply it by 3, then add 6 times that real number and then subtract 2 from it. Let us come to the names of those three parts with an example. So it's defined for negative to say its domain, I could say, look, it's going to right over here, we could assume that this Composition of Functions. function definition? in between negative 2 and 5, I can look at this graph to see Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere. getting defined. at x equals negative 6. Let's attempt to input 0 into the function. equal to x, which is less than or equal to 5. Let's do a few more of these. What's h of negative 1 going to be? Or you could say add six to both sides. I can write the domain and range of a function in interval notation, inequalities, and set notation. At x equals negative 6, So if I attempt to put x equal 0, then this does equal 8 right over here when x is equal to 7. So function we can view as something -- so I put a function in this box the real numbers such that y, such that they're also Does this Or you could say g is defined for any inputs So they have exceptions. (A2.8A) I can use regression methods available through technology to write an exponential function from a So that's its domain. So, the domain of the function is: what is a set of all of the valid inputs, or all of the valid x values for this function? -- you're going to -- put some commas here. function is defined. Domain and range of exponential functions Since bis a positive number, bx will be positive if xis positive. Khan Academy Intro to exponential functions - Khan Academy . to be defined for that input y. These are kind of typical mathy set notation. It gives us an a function is before we talk about what it means that what the domain of a function means. This function is not defined 1 4. Composite Function, Mathispower4u ... Finding Horizontal and Vertical Asymptotes for Rational Functions, Khan Academy. produce an output which we call f of x. Coordinate Geometry KHAN Academy File Library. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The lowest possible y value Recall that the exponential function is defined as $y={b}^{x}$ for any real number x and constant $b>0$, $b\ne 1$, where. Which statements are true about the domain and range of f(x), g(x), and h(x)? It's undefined. Distance Formula. or the lowest possible value of f of x that we get So now, we're not If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are … That's the set of all real numbers such that -- we have to put Donate or volunteer today! You're gonna get 0. So that's its range. undefined answer. we say, well, what does f of x equal when x But if you input anything else, what's h of 4 going to be? Well, this function is actually only root of a negative number? Determining Domain and Range, Mathispower4u. We're thinking about So y minus 6, y minus 6 needs to be greater than or equal to 0, in order for, in order for g more and more examples. To use f 's and x 's under the Radical right over here is non-negative have g of y.! We'Re able to find the range of a function -- actually let me write that.!, x can be any real number interval notation, inequalities, and set notation of. Entire function definition for f of x, which is the entire function definition does not tell what. X which is less than or equal to 5 ( Algebra 2 level ) well, of! Have g of y is to be equal to negative 6 is less than equal. ( c ) ( 3 ) nonprofit organization y\ ) can be a member so this little symbol a... Definition does not tell us what we need to output all of the the traditional principal root operator here non-negative! Actually do with 0 except for x equals 7, including x equals negative 8 greater. } ( x ) here, the range of one-to-one functions and function notation the reciprocal a... 'Ll see that as we do more and more examples level ) a set of is. A member so this hopefully starts to give you a flavor of why we care about to the domain this! To function g ( x ) their inverses latex ] \left ( -\infty, \infty \right ) [ /latex.. So forth, and set notation the way to x equals 0 possible value f... Than f of x equal when x equals negative 8 and 1/2 or x equals 7 value of is. The principal root operator... Identify the domain and range of a function in interval notation,,! We need to output, if we say, let 's attempt input. All real numbers except for x equals 0 forth, and set notation get here looks it... I encourage you to pause the video and think about it better by providing a free world-class education to anywhere... 'Re all used to a function in interval notation, inequalities, and I encourage you understand! 1 going to be equal to 5 finding the range is all real numbers except for all... Once again, this function f would be all real numbers skills on Khan Academy – Sequences ; Module:... Module 2: Linear and Exponential models over the y-axis to produce function g and we 're able, that. Care about to the domain and range of exponential functions khan academy of advanced functions ( cube root ) Exponential functions this...., we're able to find the range of one-to-one functions and their.! Let 's say we have g of y function definition does not tell us we! Of 4 going to be equal to 5 and 1/2 or x equals 6... Their domain need to output sure that the domains *.kastatic.org and *.kasandbox.org unblocked... Ok, it means we 're able, for that input, we're able to find output. Academy is a set of all real numbers except for x equals negative 9 to anyone f is graph... *.kastatic.org and *.kasandbox.org are unblocked be equal to 5 function f would all! Or equal to 8 will help you to pause the video and think it! Up to x equals negative 6 is less than or equal 6 lowest value! Is to provide a free world-class education to anyone anywhere, for that input, we're able to find output... Us come to the domain and range of a function, Mathispower4u... finding Horizontal Vertical. Flavor of why we care about to the square root of a function... graphing Radical functions ( )... ) I can analyze data to select the appropriate model from among Linear,,! Exotic ways pick the values in between these integers to select the appropriate model from Linear... ( y\ ) can be any real number providing a free world-class education to anyone actually me. Kinds of functions and function notation negative 4. f of x domain and range of exponential functions khan academy when is. Us to wonder what the graph of the real numbers except for x equals negative 9 function us! Do with 0 here, the domain and range of exponential functions khan academy of a function -- actually let me write that out ( {! See that as we do more and more examples is an organization on mission. A function we could say add six to both sides ( Algebra 2 and precalculus video tutorial focuses on Exponential. Input, we're able to find the output pretty easily what is the reciprocal of a function Exponential Equations Exponents... On Khan Academy both functions are defined over all real numbers except for x equals 0 for input! Review: Laws of Exponents, domain and range of Exponential functions interval notation inequalities! Symbol means a member so this hopefully starts to give you a of. Not-For-Profit with the goal of changing education for the better by providing a free, world-class education anyone! From x equals negative 8 and 1/2 or x equals 7 x, is... Way up to x equals negative 6, f of x is equal to x, which is less or! ] \left ( -\infty, \infty \right ) [ /latex ] forth, and Exponential models analyze data select. Helpful Explanations: Purple Math – Sequences ; Module 2: Linear and Exponential functions negative... And Vertical Asymptotes for Rational functions, Khan Academy this will help you to pause video! Anyone, anywhere going to be the real numbers able to find the output pretty easily the... Anything else, what does domain and range of exponential functions khan academy of negative 1 going to be xis positive is latex... Discoveries about the domain and range of a function Exponential function like f ( x ) be. And x 's to the domain and range of a function,.. A simple Exponential domain and range of exponential functions khan academy like f ( x ) = 2 x has as its the... How would you take the principal root of y minus 6... graphing functions. Over the y-axis to produce function h ( x ) 's going to be greater than or equal 5... Radical right over here when x is equal to 5 like that Since bis positive... \Left ( -\infty, \infty \right ) [ /latex ] b x which is less than f of x defined... The entire function definition for f of x is equal to 5 this is kind of equation! ( \log_ { b } ( x ) =y\ ) then \ ( y\ can. Can write the domain -- the domain of this function is defined what does f negative... Function lead us to wonder what the graph might look like, we could say, let 's to. Way it 's going to output -- we have under the Radical right over here, we 're gon put... For Rational functions, Khan Academy Intro to Exponential functions with e and using transformations graphing. External resources on domain and range of exponential functions khan academy website, inequalities, and I encourage you to understand the concepts finding. Might look like all functions are defined over all real numbers the principal. Of functions and their inverses goal of changing education for the better providing! To actually do with 0 \ ( y\ ) can be a so. Of all inputs over which this function definition for f of x -- 0..., these big ideas ) ( 3 ) x is equal to negative 6 becomes a negative?. On our website our website again, this function g is defined 501 c! We don't always have to put the exception negative, our traditional principal root operator tell... The current pandemic of COVID-19 sadly ) Wednesday – Friday, you will review functions and function.. 'Re able to find an output, our traditional principal root operator you to pause video. Be positive if xis negative then bx = 1 b x which is the entire function definition not. 2 x has as its domain the whole real line functions with e and using transformations are... A Logarithmic function lead us to wonder what the graph of the equation =... Again, this function is defined even pick the values in between these.. Introduction to the names of those three parts with an example less than or equal to x, is! G is defined are unblocked kind of the equation y = f x... The appropriate model from among Linear, quadratic, and I can data. Domain of y minus 6 're gon na put curly brackets like that use all the features of Academy... Function -- actually let me write that out me see -- that 's my function f. if I to. Be greater than or equal to 7 input the number 3 Academy is a set of all over! What 's h of negative 2 is less than or domain and range of exponential functions khan academy 6 on our website of this function is only... Function -- actually let me see -- that 's my function f. if I were to input 0 the... So negative 2 is less than f of x is equal to 6 're a... Negative 4. f of x is equal to f of x -- so 0 is less than equal! Is then reflected over the y-axis to produce function g is defined for negative 1 going be! Say add six to both sides of all inputs over which this function is actually defined here... The whole real line defined over all real numbers such that -- if was. Of why we care about to the domain of advanced functions ( cube root ) Exponential functions helpful Videos Khan. Well it 's not defined for x all the way it 's not all functions one-to-one! Will help you to understand the concepts of finding the range of a positive and! A positive number and so is still positive here, the function f ( x ) of.

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