Search

Text Color

Margin Size

Font Type

Enable Dyslexic Font

27.1: Review of functions and graphs

Last updated

May 2, 2022
Save as PDF
- 27: Reviews
- 27.2: Review of polynomials and rational functions

Thomas Tradler and Holly Carley
CUNY New York City College of Technology via New York City College of Technology at CUNY Academic Works

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\id}{\mathrm{id}}$ $\newcommand{\Span}{\mathrm{span}}$

( \newcommand{\kernel}{\mathrm{null}\,}\) $\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$ $\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$ $\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\id}{\mathrm{id}}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\kernel}{\mathrm{null}\,}$

$\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$

$\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$

$\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$ $\newcommand{\AA}{\unicode[.8,0]{x212B}}$

$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vectorC}[1]{\textbf{#1}}$

$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$

$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$

$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\avec}{\mathbf a}$

$\newcommand{\bvec}{\mathbf b}$

$\newcommand{\cvec}{\mathbf c}$

$\newcommand{\dvec}{\mathbf d}$

$\newcommand{\dtil}{\widetilde{\mathbf d}}$

$\newcommand{\evec}{\mathbf e}$

$\newcommand{\fvec}{\mathbf f}$

$\newcommand{\nvec}{\mathbf n}$

$\newcommand{\pvec}{\mathbf p}$

$\newcommand{\qvec}{\mathbf q}$

$\newcommand{\svec}{\mathbf s}$

$\newcommand{\tvec}{\mathbf t}$

$\newcommand{\uvec}{\mathbf u}$

$\newcommand{\vvec}{\mathbf v}$

$\newcommand{\wvec}{\mathbf w}$

$\newcommand{\xvec}{\mathbf x}$

$\newcommand{\yvec}{\mathbf y}$

$\newcommand{\zvec}{\mathbf z}$

$\newcommand{\rvec}{\mathbf r}$

$\newcommand{\mvec}{\mathbf m}$

$\newcommand{\zerovec}{\mathbf 0}$

$\newcommand{\onevec}{\mathbf 1}$

$\newcommand{\real}{\mathbb R}$

$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$

$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$

$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$

$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$

$\newcommand{\bcal}{\cal B}$

$\newcommand{\ccal}{\cal C}$

$\newcommand{\scal}{\cal S}$

$\newcommand{\wcal}{\cal W}$

$\newcommand{\ecal}{\cal E}$

$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$

$\newcommand{\gray}[1]{\color{gray}{#1}}$

$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$

$\newcommand{\rank}{\operatorname{rank}}$

$\newcommand{\row}{\text{Row}}$

$\newcommand{\col}{\text{Col}}$

$\renewcommand{\row}{\text{Row}}$

$\newcommand{\nul}{\text{Nul}}$

$\newcommand{\var}{\text{Var}}$

$\newcommand{\corr}{\text{corr}}$

$\newcommand{\len}[1]{\left|#1\right|}$

$\newcommand{\bbar}{\overline{\bvec}}$

$\newcommand{\bhat}{\widehat{\bvec}}$

$\newcommand{\bperp}{\bvec^\perp}$

$\newcommand{\xhat}{\widehat{\xvec}}$

$\newcommand{\vhat}{\widehat{\vvec}}$

$\newcommand{\uhat}{\widehat{\uvec}}$

$\newcommand{\what}{\widehat{\wvec}}$

$\newcommand{\Sighat}{\widehat{\Sigma}}$

$\newcommand{\lt}{<}$

$\newcommand{\gt}{>}$

$\newcommand{\amp}{&}$

$\definecolor{fillinmathshade}{gray}{0.9}$

Exercise $\PageIndex{1}$

Find all solutions of the equation: $\big| 3x-9\big| =6$

Answer: $x = 1$ or $x = 5$

Exercise $\PageIndex{2}$

Find the equation of the line displayed below.

$clipboard_e6193cbb38730a86463aff4c9f1015643.png$

Answer: $y=-\dfrac{1}{2} x+1$

Exercise $\PageIndex{3}$

Find the solution of the equation: $x^3-3x^2+2x-2=0$ Approximate your answer to the nearest hundredth.

Answer: $x \approx 2.52$

Exercise $\PageIndex{4}$

Let $f(x)=x^2-2x+5$ . Simplify the difference quotient $\dfrac{f(x+h)-f(x)}{h}$ as much as possible.

Answer: $2 x-2+2 h$

Exercise $\PageIndex{5}$

Consider the following graph of a function $f$ .

$clipboard_ebbf8b3d341de54a74f708abe18207254.png$

Find: domain of $f$ , range of $f$ , $f(3)$ , $f(5)$ , $f(7)$ , $f(9)$ .

Answer: domain $D=[2,7]$ , range $R=(1,4], f(3)=2, f(5)=2, f(7)=4, f(9)$ is undefined

Exercise $\PageIndex{6}$

Find the formula of the graph displayed below.

$clipboard_e79aaccb6dd8116b811f41f51239d7955.png$

Answer: $f(x)=-x^{2}+2$

Exercise $\PageIndex{7}$

Let $f(x)=5x+4$ and $g(x)=x^2+8x+7$ . Find the quotient $\left(\dfrac f g\right)(x)$ and state its domain.

Answer: $\left(\dfrac{f}{g}\right)(x)=\dfrac{5 x+4}{x^{2}+8 x+7}=\dfrac{5 x+4}{(x+7)(x+1)}$ has domain $D=\mathbb{R}-\{-7,-1\}$

Exercise $\PageIndex{8}$

Let $f(x)=x^2+\sqrt{x-3}$ and $g(x)=2x-3$ . Find the composition $(f\circ g)(x)$ and state its domain.

Answer: $(f \circ g)(x)=4 x^{2}-12 x+9+\sqrt{2 x-6}$ has domain $D=[3, \infty)$

Exercise $\PageIndex{9}$

Consider the assignments for $f$ and $g$ given by the table below.

$\begin{array}{|c||c|c|c|c|c|} \hline x & 2 & 3 & 4 & 5 & 6 \\ \hline f(x) & 5 & 0 & 2 & 4 & 2 \\ \hline g(x) & 6 & 2 & 3 & 4 & 1 \\ \hline \end{array} \nonumber$

Is $f$ a function? Is $g$ a function? Write the composed assignment for $(f\circ g)(x)$ as a table.

Answer

$f$ and $g$ are both functions, and the composition is given by the table:

$\begin{array}{|c||c|c|c|c|c|} \hline x & 2 & 3 & 4 & 5 & 6 \\ \hline \hline f(x) & 5 & 0 & 2 & 4 & 2 \\ \hline g(x) & 6 & 2 & 3 & 4 & 1 \\ \hline(f \circ g)(x) & 2 & 5 & 0 & 2 & \text { undef. } \\ \hline \end{array}$

Exercise $\PageIndex{10}$

Find the inverse of the function $f(x)=\dfrac{1}{2x+5}$ .

Answer: $f^{-1}(x)=\dfrac{1}{2 x}-\dfrac{5}{2}$

Exercise 27.1.1\PageIndex{1}

Exercise 27.1.2\PageIndex{2}

Exercise 27.1.3\PageIndex{3}

Exercise 27.1.4\PageIndex{4}

Exercise 27.1.5\PageIndex{5}

Exercise 27.1.6\PageIndex{6}

Exercise 27.1.7\PageIndex{7}

Exercise 27.1.8\PageIndex{8}

Exercise 27.1.9\PageIndex{9}

Exercise 27.1.10\PageIndex{10}

Support Center

How can we help?