Properties of Functions

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\(\def\vs{{\it vs. }} \def\cf{{\it cf. }} \def\viz{{\it viz. }} \def\ie{{\it i.e. }} \def\etc{{\it etc. }} \def\eg{{\it e.g. }} \def\etal{{\it et al .}} \def\via{{\it via }} \def\adhoc{{\it ad hoc }} \def\apriori{{\it apriori }} \def\Afrak{\mathfrak{A}} \def\Bfrak{\mathfrak{B}} \def\Cfrak{\mathfrak{C}} \def\Dfrak{\mathfrak{D}} \def\Efrak{\mathfrak{E}} \def\Ffrak{\mathfrak{F}} \def\Gfrak{\mathfrak{G}} \def\Hfrak{\mathfrak{H}} \def\Ifrak{\mathfrak{I}} \def\Jfrak{\mathfrak{J}} \def\Kfrak{\mathfrak{K}} \def\Lfrak{\mathfrak{L}} \def\Mfrak{\mathfrak{M}} \def\Nfrak{\mathfrak{N}} \def\Ofrak{\mathfrak{O}} \def\Pfrak{\mathfrak{P}} \def\Qfrak{\mathfrak{Q}} \def\Rfrak{\mathfrak{R}} \def\Sfrak{\mathfrak{S}} \def\Tfrak{\mathfrak{T}} \def\Ufrak{\mathfrak{U}} \def\Vfrak{\mathfrak{V}} \def\Wfrak{\mathfrak{W}} \def\Xfrak{\mathfrak{X}} \def\Yfrak{\mathfrak{Y}} \def\Zfrak{\mathfrak{Z}} \def\afrak{\mathfrak{a}} \def\bfrak{\mathfrak{b}} \def\cfrak{\mathfrak{c}} \def\dfrak{\mathfrak{d}} \def\efrak{\mathfrak{e}} \def\ffrak{\mathfrak{f}} \def\gfrak{\mathfrak{g}} \def\hfrak{\mathfrak{h}} \def\ifrak{\mathfrak{i}} \def\jfrak{\mathfrak{j}} \def\kfrak{\mathfrak{k}} \def\lfrak{\mathfrak{l}} \def\mfrak{\mathfrak{m}} \def\nfrak{\mathfrak{n}} \def\ofrak{\mathfrak{o}} \def\pfrak{\mathfrak{p}} \def\qfrak{\mathfrak{q}} \def\rfrak{\mathfrak{r}} \def\sfrak{\mathfrak{s}} \def\tfrak{\mathfrak{t}} \def\ufrak{\mathfrak{u}} \def\vfrak{\mathfrak{v}} \def\wfrak{\mathfrak{w}} \def\xfrak{\mathfrak{x}} \def\yfrak{\mathfrak{y}} \def\zfrak{\mathfrak{z}} \def\AA{\mathbb{A}} \def\BB{\mathbb{B}} \def\CC{\mathbb{C}} \def\DD{\mathbb{D}} \def\EE{\mathbb{E}} \def\FF{\mathbb{F}} \def\GG{\mathbb{G}} \def\HH{\mathbb{H}} \def\II{\mathbb{I}} \def\JJ{\mathbb{J}} \def\KK{\mathbb{K}} \def\LL{\mathbb{L}} \def\MM{\mathbb{M}} \def\NN{\mathbb{N}} \def\OO{\mathbb{O}} \def\PP{\mathbb{P}} \def\QQ{\mathbb{Q}} \def\RR{\mathbb{R}} \def\SS{\mathbb{S}} \def\TT{\mathbb{T}} \def\UU{\mathbb{U}} \def\VV{\mathbb{V}} \def\WW{\mathbb{W}} \def\XX{\mathbb{X}} \def\YY{\mathbb{Y}} \def\ZZ{\mathbb{Z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\Ap{A^\prime} \def\Bp{B^\prime} \def\Cp{C^\prime} \def\Dp{D^\prime} \def\Ep{E^\prime} \def\Fp{F^\prime} \def\Gp{G^\prime} \def\Hp{H^\prime} \def\Ip{I^\prime} \def\Jp{J^\prime} \def\Kp{K^\prime} \def\Lp{L^\prime} \def\Mp{M^\prime} \def\Mp{N^\prime} \def\Op{O^\prime} \def\Pp{P^\prime} \def\Qp{Q^\prime} \def\Rp{R^\prime} \def\Sp{S^\prime} \def\Tp{T^\prime} \def\Up{U^\prime} \def\Vp{V^\prime} \def\Wp{W^\prime} \def\Xp{X^\prime} \def\Yp{Y^\prime} \def\Zp{Z^\prime} \def\ap{a^\prime} \def\bp{b^\prime} \def\cp{c^\prime} \def\dprime{d^\prime} \def\ep{e^\prime} \def\fp{f^\prime} \def\gp{g^\prime} \def\hp{h^\prime} \def\ip{i^\prime} \def\jp{j^\prime} \def\kp{k^\prime} \def\lp{l^\prime} \def\mp{m^\prime} \def\np{n^\prime} \def\op{o^\prime} \def\pp{p^\prime} \def\qp{q^\prime} \def\rp{r^\prime} \def\sp{s^\prime} \def\tp{t^\prime} \def\up{u^\prime} \def\vp{v^\prime} \def\wp{w^\prime} \def\xp{x^\prime} \def\yp{y^\prime} \def\zp{z^\prime} \def\App{A^{\prime\prime}} \def\Bpp{B^{\prime\prime}} \def\Cpp{C^{\prime\prime}} \def\Dpp{D^{\prime\prime}} \def\Epp{E^{\prime\prime}} \def\Fpp{F^{\prime\prime}} \def\Gpp{G^{\prime\prime}} \def\Hpp{H^{\prime\prime}} \def\Ipp{I^{\prime\prime}} \def\Jpp{J^{\prime\prime}} \def\Kpp{K^{\prime\prime}} \def\Lpp{L^{\prime\prime}} \def\Mpp{M^{\prime\prime}} \def\Mpp{N^{\prime\prime}} \def\Opp{O^{\prime\prime}} \def\Ppp{P^{\prime\prime}} \def\Qpp{Q^{\prime\prime}} \def\Rpp{R^{\prime\prime}} \def\Spp{S^{\prime\prime}} \def\Tpp{T^{\prime\prime}} \def\Upp{U^{\prime\prime}} \def\Vpp{V^{\prime\prime}} \def\Wpp{W^{\prime\prime}} \def\Xpp{X^{\prime\prime}} \def\Ypp{Y^{\prime\prime}} \def\Zpp{Z^{\prime\prime}} \def\app{a^{\prime\prime}} \def\bpp{b^{\prime\prime}} \def\cpp{c^{\prime\prime}} \def\dpp{d^{\prime\prime}} \def\epp{e^{\prime\prime}} \def\fpp{f^{\prime\prime}} \def\gpp{g^{\prime\prime}} \def\hpp{h^{\prime\prime}} \def\ipp{i^{\prime\prime}} \def\jpp{j^{\prime\prime}} \def\kpp{k^{\prime\prime}} \def\lpp{l^{\prime\prime}} \def\mpp{m^{\prime\prime}} \def\npp{n^{\prime\prime}} \def\opp{o^{\prime\prime}} \def\ppp{p^{\prime\prime}} \def\qpp{q^{\prime\prime}} \def\rpp{r^{\prime\prime}} \def\spp{s^{\prime\prime}} \def\tpp{t^{\prime\prime}} \def\upp{u^{\prime\prime}} \def\vpp{v^{\prime\prime}} \def\wpp{w^{\prime\prime}} \def\xpp{x^{\prime\prime}} \def\ypp{y^{\prime\prime}} \def\zpp{z^{\prime\prime}} \def\abar{\overline{a}} \def\bbar{\overline{b}} \def\cbar{\overline{c}} \def\dbar{\overline{d}} \def\ebar{\overline{e}} \def\fbar{\overline{f}} \def\gbar{\overline{g}} \def\ibar{\overline{i}} \def\jbar{\overline{j}} \def\kbar{\overline{k}} \def\lbar{\overline{l}} \def\mbar{\overline{m}} \def\nbar{\overline{n}} \def\obar{\overline{o}} \def\pbar{\overline{p}} \def\qbar{\overline{q}} \def\rbar{\overline{r}} \def\sbar{\overline{s}} \def\tbar{\overline{t}} \def\ubar{\overline{u}} \def\vbar{\overline{v}} \def\wbar{\overline{w}} \def\xbar{\overline{x}} \def\ybar{\overline{y}} \def\zbar{\overline{z}} \def\Abar{\overline{A}} \def\Bbar{\overline{B}} \def\Cbar{\overline{C}} \def\Dbar{\overline{D}} \def\Ebar{\overline{E}} \def\Fbar{\overline{F}} \def\Gbar{\overline{G}} \def\Hbar{\overline{H}} \def\Ibar{\overline{I}} \def\Jbar{\overline{J}} \def\Kbar{\overline{K}} \def\Lbar{\overline{L}} \def\Mbar{\overline{M}} \def\Nbar{\overline{N}} \def\Obar{\overline{O}} \def\Pbar{\overline{P}} \def\Qbar{\overline{Q}} \def\Rbar{\overline{R}} \def\Sbar{\overline{S}} \def\Tbar{\overline{T}} \def\Ubar{\overline{U}} \def\Vbar{\overline{V}} \def\Wbar{\overline{W}} \def\Xbar{\overline{X}} \def\Ybar{\overline{Y}} \def\Zbar{\overline{Z}} \def\aunder{\underline{a}} \def\bunder{\underline{b}} \def\cunder{\underline{c}} \def\dunder{\underline{d}} \def\eunder{\underline{e}} \def\funder{\underline{f}} \def\gunder{\underline{g}} \def\hunder{\underline{h}} \def\iunder{\underline{i}} \def\junder{\underline{j}} \def\kunder{\underline{k}} \def\lunder{\underline{l}} \def\munder{\underline{m}} \def\nunder{\underline{n}} \def\ounder{\underline{o}} \def\punder{\underline{p}} \def\qunder{\underline{q}} \def\runder{\underline{r}} \def\sunder{\underline{s}} \def\tunder{\underline{t}} \def\uunder{\underline{u}} \def\vunder{\underline{v}} \def\wunder{\underline{w}} \def\xunder{\underline{x}} \def\yunder{\underline{y}} \def\zunder{\underline{z}} \def\Aunder{\underline{A}} \def\atilde{\widetilde{a}} \def\btilde{\widetilde{b}} \def\ctilde{\widetilde{c}} \def\dtilde{\widetilde{d}} \def\etilde{\widetilde{e}} \def\ftilde{\widetilde{f}} \def\gtilde{\widetilde{g}} \def\htilde{\widetilde{h}} \def\itilde{\widetilde{i}} \def\jtilde{\widetilde{j}} \def\ktilde{\widetilde{k}} \def\ltilde{\widetilde{l}} \def\mtilde{\widetilde{m}} \def\ntilde{\widetilde{n}} \def\otilde{\widetilde{o}} \def\ptilde{\widetilde{p}} \def\qtilde{\widetilde{q}} \def\rtilde{\widetilde{r}} \def\stilde{\widetilde{s}} \def\ttilde{\widetilde{t}} \def\utilde{\widetilde{u}} \def\vtilde{\widetilde{v}} \def\wtilde{\widetilde{w}} \def\xtilde{\widetilde{x}} \def\ytilde{\widetilde{y}} \def\ztilde{\widetilde{z}} \def\Atilde{\widetilde{A}} \def\Btilde{\widetilde{B}} \def\Ctilde{\widetilde{C}} \def\Dtilde{\widetilde{D}} \def\Etilde{\widetilde{E}} \def\Ftilde{\widetilde{F}} \def\Gtilde{\widetilde{G}} \def\Htilde{\widetilde{H}} \def\Itilde{\widetilde{I}} \def\Jtilde{\widetilde{J}} \def\Ktilde{\widetilde{K}} \def\Ltilde{\widetilde{L}} \def\Mtilde{\widetilde{M}} \def\Ntilde{\widetilde{N}} \def\Otilde{\widetilde{O}} \def\Ptilde{\widetilde{P}} \def\Qtilde{\widetilde{Q}} \def\Rtilde{\widetilde{R}} \def\Stilde{\widetilde{S}} \def\Ttilde{\widetilde{T}} \def\Utilde{\widetilde{U}} \def\Vtilde{\widetilde{V}} \def\Wtilde{\widetilde{W}} \def\Xtilde{\widetilde{X}} \def\Ytilde{\widetilde{Y}} \def\Ztilde{\widetilde{Z}} \def\Alphatilde{\widetilde{\Alpha}} \def\Betatilde{\widetilde{\Beta}} \def\Gammatilde{\widetilde{\Gamma}} \def\Deltatilde{\widetilde{\Delta}} \def\Epsilontilde{\widetilde{\Epsilon}} \def\Zetatilde{\widetilde{\Zeta}} \def\Etatilde{\widetilde{\Eta}} \def\Thetatilde{\widetilde{\Theta}} \def\Iotatilde{\widetilde{\Iota}} \def\Kappatilde{\widetilde{\Kappa}} \def\Lambdatilde{\widetilde{\Lamdba}} \def\Mutilde{\widetilde{\Mu}} \def\Nutilde{\widetilde{\Nu}} \def\Xitilde{\widetilde{\Xi}} \def\Omicrontilde{\widetilde{\Omicron}} \def\Pitilde{\widetilde{\Pi}} \def\Rhotilde{\widetilde{\Rho}} \def\Sigmatilde{\widetilde{\Sigma}} \def\Tautilde{\widetilde{\Tau}} \def\Upsilontilde{\widetilde{\Upsilon}} \def\Phitilde{\widetilde{\Phi}} \def\Chitilde{\widetilde{\Chi}} \def\Psitilde{\widetilde{\Psi}} \def\Omegatilde{\widetilde{\Omega}} \def\alphatilde{\widetilde{\alpha}} \def\betatilde{\widetilde{\beta}} \def\gammatilde{\widetilde{\gamma}} \def\deltatilde{\widetilde{\delta}} \def\epsilontilde{\widetilde{\epsilon}} \def\zetatilde{\widetilde{\zeta}} \def\etatilde{\widetilde{\eta}} \def\thetatilde{\widetilde{\theta}} \def\iotatilde{\widetilde{\iota}} \def\kappatilde{\widetilde{\kappa}} \def\lambdatilde{\widetilde{\lamdba}} \def\mutilde{\widetilde{\mu}} \def\nutilde{\widetilde{\nu}} \def\xitilde{\widetilde{\xi}} \def\omicrontilde{\widetilde{\omicron}} \def\pitilde{\widetilde{\pi}} \def\rhotilde{\widetilde{\rho}} \def\sigmatilde{\widetilde{\sigma}} \def\tautilde{\widetilde{\tau}} \def\upsilontilde{\widetilde{\upsilon}} \def\phitilde{\widetilde{\phi}} \def\chitilde{\widetilde{\chi}} \def\psitilde{\widetilde{\psi}} \def\omegatilde{\widetilde{\omega}} \def\Alphabar{\bar{\Alpha}} \def\Betabar{\bar{\Beta}} \def\Gammabar{\bar{\Gamma}} \def\Deltabar{\bar{\Delta}} \def\Epsilonbar{\bar{\Epsilon}} \def\Zetabar{\bar{\Zeta}} \def\Etabar{\bar{\Eta}} \def\Thetabar{\bar{\Theta}} \def\Iotabar{\bar{\Iota}} \def\Kappabar{\bar{\Kappa}} \def\Lambdabar{\bar{\Lamdba}} \def\Mubar{\bar{\Mu}} \def\Nubar{\bar{\Nu}} \def\Xibar{\bar{\Xi}} \def\Omicronbar{\bar{\Omicron}} \def\Pibar{\bar{\Pi}} \def\Rhobar{\bar{\Rho}} \def\Sigmabar{\bar{\Sigma}} \def\Taubar{\bar{\Tau}} \def\Upsilonbar{\bar{\Upsilon}} \def\Phibar{\bar{\Phi}} \def\Chibar{\bar{\Chi}} \def\Psibar{\bar{\Psi}} \def\Omegabar{\bar{\Omega}} \def\alphabar{\bar{\alpha}} \def\betabar{\bar{\beta}} \def\gammabar{\bar{\gamma}} \def\deltabar{\bar{\delta}} \def\epsilonbar{\bar{\epsilon}} \def\zetabar{\bar{\zeta}} \def\etabar{\bar{\eta}} \def\thetabar{\bar{\theta}} \def\iotabar{\bar{\iota}} \def\kappabar{\bar{\kappa}} \def\lambdabar{\bar{\lamdba}} \def\mubar{\bar{\mu}} \def\nubar{\bar{\nu}} \def\xibar{\bar{\xi}} \def\omicronbar{\bar{\omicron}} \def\pibar{\bar{\pi}} \def\rhobar{\bar{\rho}} \def\sigmabar{\bar{\sigma}} \def\taubar{\bar{\tau}} \def\upsilonbar{\bar{\upsilon}} \def\phibar{\bar{\phi}} \def\chibar{\bar{\chi}} \def\psibar{\bar{\psi}} \def\omegabar{\bar{\omega}} \def\del{\partial} \def\delbar{\overline{\partial}} \def\Cech{\check{C}} \def\half{\frac{1}{2}} \def\defeq{\mathrel{\mathop:}=} \def\alg{\mathrm{alg}} \def\Alt{\mathrm{Alt}} \def\Amp{\mathrm{Amp}} \def\Arg{\mathrm{Arg}} \def\an{\mathrm{an}} \def\anti{\mathrm{anti}} \def\Ap{\mathrm{Ap}} \def\arcsinh{\mathrm{arcsinh\hspace{0.07cm}}} \def\arccosh{\mathrm{arccosh\hspace{0.07cm}}} \def\arctanh{\mathrm{arctanh\hspace{0.07cm}}} \def\arccsch{\mathrm{arccsch\hspace{0.07cm}}} \def\arcsech{\mathrm{arcsech\hspace{0.07cm}}} \def\arccoth{\mathrm{arccoth\hspace{0.07cm}}} \def\arccsc{\mathrm{arccsc\hspace{0.07cm}}} \def\arcsec{\mathrm{arcsec\hspace{0.07cm}}} \def\arccot{\mathrm{arccot\hspace{0.07cm}}} \def\arg{\mathrm{arg}} \def\BC{\mathrm{BC}} \def\Bel{\mathrm{Bel}} \def\calCH{\mathcal{CH}} \def\csch{\mathrm{csch}\hspace{0.07cm}} \def\CH{\mathrm{CH}} \def\ch{\mathrm{ch}} \def\closed{\mathrm{closed}} \def\codim{\mathrm{codim}} \def\coth{\mathrm{coth}\hspace{0.07cm}} \def\Coh{\mathfrak{Coh}} \def\Coker{\mathrm{Coker}} \def\Cone{\mathrm{Cone}} \def\darg{d\mathrm{arg}} \def\Db{\mathrm{Db}} \def\dclosed{\mathrm{d-closed}} \def\deg{\mathrm{deg}} \def\dim{\mathrm{dim}} \def\divisor{\mathrm{div}} \def\dlog{d\mathrm{log}} \def\DNE{\mathrm{DNE}} \def\DR{\mathrm{DR}} \def\DST{\mathrm{DST}} \def\exp{\mathrm{exp}} \def\FLB{\mathrm{FLB}} \def\FLS{\mathrm{FLS}} \def\Gr{\mathrm{Gr}} \def\Hzar{H_{\mathrm{Zar}}} \def\Hol{\mathrm{Hol}} \def\Id{\mathrm{Id}} \def\Image{\mathrm{Im}} \def\Ka{\mathcal{K}_A} \def\Ker{\mathrm{Ker}} \def\kod{\mathrm{kod}} \def\Kx{\mathcal{K}_X} \def\Kz{\mathcal{K}_Z} \def\log{\mathrm{log}} \def\Log{\mathrm{Log}} \def\Li{\mathrm{Li}} \def\min{\mathrm{min}} \def\Mon{\mathrm{Mon}} \def\Nef{\mathrm{Nef}} \def\NS{\mathrm{NS}} \def\Oa{\mathcal{O}_A} \def\Ox{\mathcal{O}_X} \def\Oz{\mathcal{O}_Z} \def\Perp{\mathrm{Perp}} \def\Pic{\mathrm{Pic}} \def\Proj{\mathrm{Proj}} \def\rank{\mathrm{rank}} \def\Rat{\mathrm{Rat}} \def\Real{\mathrm{Re}} \def\reg{\mathrm{reg}} \def\Res{\mathrm{Res}} \def\res{\mathrm{res}} \def\Ric{\mathrm{Ric}} \def\sech{\mathrm{sech}\hspace{0.07cm}} \def\Span{\mathrm{Span}} \def\Spec{\mathrm{Spec}} \def\sing{\mathrm{sing}} \def\Singx{\mathrm{Sing}(X)} \def\sheafKer{\mathcal{\Ker}} \def\sheafIm{\mathcal{\Im}} \def\Span{\mathrm{Span}} \def\Spin{\mathrm{Spin}} \def\Str{\mathrm{Str}} \def\td{\mathrm{td}} \def\tr{\mathrm{tr}} \def\Todd{\mathrm{Todd}} \def\tor{\mathrm{tor}} \def\trdeg{\mathrm{trdeg}} \def\Zar{\mathrm{Zar}} \def\ZFLS{\mathrm{ZFLS}} \usepackage{tikz} \usepackage{tkz-graph} \usepackage{tkz-euclide} \usetikzlibrary{patterns} \usetikzlibrary{positioning} \usetikzlibrary{matrix,arrows} \usetikzlibrary{calc} \usetikzlibrary{shapes} \usetikzlibrary{through,intersections,decorations,shadows,fadings} \usepackage{pgfplots} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} \)

Section 2.3 Properties of Functions

Subsection 2.3.1 First Properties

Calculus studies the behaviour of functions. Many behaviours are described by defining various properties. Many of the properties defined here may be review from your mathematical experience, though the terms may have changed.

The domain of a function is all possible (real number) inputs to the function. The domain can be restricted to a smaller subset if desired.
The range of a function is all possible (real number) outputs of the function.
A function is even if \(f(-x) = f(x)\text{.}\) Visually, the graph of the function has a mirror-image symmetry over the \(y\) axis.
A function is odd if \(f(-x) = -f(x)\text{.}\) Visually, the graph of the function is the preserved under a half-turn rotation about the origin.
A function is periodic if there is a real number \(a\) such that \(f(x+a) = f(a)\) for all \(x\text{.}\) The smallest positive real number \(a\) that satisfies is called the period.
A function is called increasing if \(b \gt a\) implies \(f(b) \gt f(a)\text{.}\) Visually, the graph of the function is growing upwards as the input increases.
A function is called decreasing if \(b \gt a\) implies \(f(b) \lt f(a)\text{.}\) Visually, the graph of the function is declining downwards as the input increases.
A function is called monotonic if it is either always increasing or always decreasing.
A function is bounded above if there is a number \(A\) such that \(f(x) \lt A\) for all \(x\text{.}\) Visually, the graph of the function never gets above the height \(y=A\text{.}\)
A function is bounded below if there is a number \(B\) such that \(f(x) > B\) for all \(x\text{.}\) Visually, the graph of the function never gets below the height \(y=B\text{.}\)
A function is bounded if it is both bounded above and below. That is, there exists numbers \(A\) and \(B\) such that \(B \lt f(x) \lt A\) for all \(x\text{.}\) Visually, the function remains within the range \(y \in [A,B]\) for any input.
A \(y\)-intercept of a function is a place where it crosses the \(y\) axis. Since a function satisfies a vertical line test, there is only one possible \(y\) intercept found at \(f(0)\) (if \(0\) is in the domain).
An \(x\)-intercept or root of a function is a place where it crosses the \(x\) axis or, equivalently, has the output \(f(x) = 0\text{.}\)