Section 1
Inverse Functions
Given the function $f=\{(1,2),(2,3),(3,4)\}$ write the set of ordered pairs representing $f^{-1}$.
A function $f$ is a _________________ - _______________________ - ___________________ function if for $a$ and $b$ in the domain of $f$, if $a \neq b$, then $f(a) \neq f(b)$ .
The graph of a function and its inverse are symmetric with respect to the line ____________.
The function $f=\{(1,5),(-2,3),(-4,2),(2,5)\}$ (is/is not) a one-to-one function.
A function defined by $y=f(x)$ (is/is not) a one-to-one function if no horizontal line intersects the graph of $f$ in more than one point.
Given a one-to-one function defined by $y=f(x)$, if $f(a)=f(b),$ then $a$ ____________, b.
Let $f$ be a one-to-one function and let $g$ be the inverse of $f$. Then $(f \circ g)(x)=$ ___________ and $(g \circ f)(x)=$ _______________.
The notation ______________ is often used to represent the inverse of a function $f$ and not the reciprocal of $f$.
If $(a, b)$ is a point on the graph of a one-to-one function $f,$ then the corresponding ordered pair ___________is a point on the graph of $f^{-1}$.
The function defined by $f(x)=x^{2}-9$ (is/is not) a one-to-one function, whereas $g(x)=x^{2}-9 ; x \geq 0$ (is/is not) a one-to-one function.
A relation in $x$ and $y$ is given. Determine if the relation defines $y$ as a one-to-one function of $x$.$$\{(6,-5),(4,2),(3,1),(8,4)\}$$
A relation in $x$ and $y$ is given. Determine if the relation defines $y$ as a one-to-one function of $x$.$$\{(-14,1),(-2,3),(7,4),(-9,-2)\}$$
A relation in $x$ and $y$ is given. Determine if the relation defines $y$ as a one-to-one function of $x$.$$\begin{array}{|c|c|c|}\hline & A & B \\\hline 1 & x & y \\\hline 2 & 0.6 & 1.8 \\\hline 3 & 1 & -1.1 \\\hline 4 & 0.5 & 1.8 \\\hline 5 & 2.4 & 0.7 \\\hline\end{array}$$
A relation in $x$ and $y$ is given. Determine if the relation defines $y$ as a one-to-one function of $x$.$$\begin{array}{|c|c|c|}\hline & A & B \\\hline 1 & x & y \\\hline 2 & 12.5 & 3.21 \\\hline 3 & 5.75 & -4.5 \\\hline 4 & 2.34 & 7.25 \\\hline 5 & -12.7 & 3.21 \\\hline\end{array}$$
A relation in $x$ and $y$ is given. Determine if the relation defines $y$ as a one-to-one function of $x$.$$\begin{array}{|c|c|}\hline \boldsymbol{x} & \boldsymbol{y} \\\hline \text { California } & \text { Sacramento } \\\hline \text { Texas } & \text { Austin } \\\hline \text { New York } & \text { Albany } \\\hline \text { Florida } & \text { Tallahassee } \\\hline\end{array}$$
A relation in $x$ and $y$ is given. Determine if the relation defines $y$ as a one-to-one function of $x$.$$\begin{array}{|c|c|}\hline x & y \\\hline \text { Green Bay } & \text { Packers } \\\hline \text { Miami } & \text { Dolphins } \\\hline \text { Atlanta } & \text { Falcons } \\\hline \text { Denver } & \text { Broncos } \\\hline\end{array}$$
A relation in $x$ and $y$ is given. Determine if the relation defines $y$ as a one-to-one function of $x$.
Determine if the relation defines $y$ as a one-to-one function of $x$.
Use the definition of a one-to-one function to determine if the function is one-to-one.$f(x)=4 x-7$
Use the definition of a one-to-one function to determine if the function is one-to-one. $h(x)=-3 x+2$
Use the definition of a one-to-one function to determine if the function is one-to-one. $g(x)=x^{3}+8$
Use the definition of a one-to-one function to determine if the function is one-to-one.$k(x)=x^{3}-27$
Use the definition of a one-to-one function to determine if the function is one-to-one.$m(x)=x^{2}-4$
Use the definition of a one-to-one function to determine if the function is one-to-one.$n(x)=x^{2}+1$
Use the definition of a one-to-one function to determine if the function is one-to-one.$p(x)=|x+1|$
Use the definition of a one-to-one function to determine if the function is one-to-one. $q(x)=|x-3|$
Determine whether the two functions are inverses.$f(x)=5 x+4$ and $g(x)=\frac{x-4}{5}$
Determine whether the two functions are inverses.$h(x)=7 x-3$ and $k(x)=\frac{x+3}{7}$
Determine whether the two functions are inverses. $m(x)=\frac{-2+x}{6}$ and $n(x)=6 x-2$
Determine whether the two functions are inverses. $p(x)=\frac{-3+x}{4}$ and $q(x)=4 x-3$
Determine whether the two functions are inverses. $t(x)=\frac{4}{x-1}$ and $v(x)=\frac{x+4}{x}$
Determine whether the two functions are inverses. $w(x)=\frac{6}{x+2}$ and $z(x)=\frac{6-2 x}{x}$
There were 2000 applicants for enrollment to the freshman class at a small college in the year $2010 .$ The number of applications has risen linearly by roughly 150 per year. The number of applications $f(x)$ is given by $f(x)=2000+150 x,$ where $x$ is the number of years since 2010 .a. Determine if the function $g(x)=\frac{x-2000}{150}$ is the inverse of $f$b. Interpret the meaning of function $g$ in the context of this problem.
The monthly sales for January for a whole foods market was $\$ 60,000$ and has increased linearly by $\$ 2500$ per month. The amount in sales $f(x)$ (in \$) is given by $f(x)=60,000+2500 x$, where $x$ is the number of months since January.a. Determine if the function $g(x)=\frac{x-60,000}{2500}$ is the inverse of $f$.b. Interpret the meaning of function $g$ in the context of this problem.
a. Show that $f(x)=2 x-3$ defines a one-to-one function.b. Write an equation for $f^{1}(x)$.c. Graph $y=f(x)$ and $y=f^{-1}(x)$ on the same coordinate system.
a. Show that $f(x)=4 x+4$ defines a one-to-one function.b. Write an equation for $f^{1}(x)$.c. Graph $y=f(x)$ and $y=f^{-1}(x)$ on the same coordinate system.
A one-to-one function is given. Write an equation for the inverse function. $f(x)=\frac{4-x}{9}$
A one-to-one function is given. Write an equation for the inverse function. $g(x)=\frac{8-x}{3}$
A one-to-one function is given. Write an equation for the inverse function.$h(x)=\sqrt[3]{x-5}$
A one-to-one function is given. Write an equation for the inverse function. $k(x)=\sqrt[3]{x+8}$
A one-to-one function is given. Write an equation for the inverse function. $m(x)=4 x^{3}+2$
A one-to-one function is given. Write an equation for the inverse function. $n(x)=2 x^{3}-5$
A one-to-one function is given. Write an equation for the inverse function.$c(x)=\frac{5}{x+2}$
A one-to-one function is given. Write an equation for the inverse function. $s(x)=\frac{2}{x-3}$
A one-to-one function is given. Write an equation for the inverse function.$t(x)=\frac{x-4}{x+2}$
A one-to-one function is given. Write an equation for the inverse function.$v(x)=\frac{x-5}{x+1}$
A one-to-one function is given. Write an equation for the inverse function. $f(x)=\frac{(x-a)^{3}}{b}-c$
A one-to-one function is given. Write an equation for the inverse function.$g(x)=b(x+a)^{3}+c$
a. Graph $f(x)=x^{2}-3 ; x \leq 0 .$ (See Example 7)b. From the graph of $f$, is $f$ a one-to-one function?c. Write the domain of $f$ in interval notation.d. Write the range of $f$ in interval notation.e. Write an equation for $f^{1}(x)$.f. Graph $y=f(x)$ and $y=f^{-1}(x)$ on the same coordinate system.g. Write the domain of $f^{-1}$ in interval notation.h. Write the range of $f^{-1}$ in interval notation.
a. Graph $f(x)=x^{2}+1 ; x \leq 0$.b. From the graph of $f,$ is $f$ a one-to-one function?c. Write the domain of $f$ in interval notation.d. Write the range of $f$ in interval notation.e. Write an equation for $f^{1}(x)$.f. Graph $y=f(x)$ and $y=f^{-1}(x)$ on the same coordinate system.g. Write the domain of $f^{-1}$ in interval notation.h. Write the range of $f^{-1}$ in interval notation.
a. Graph $f(x)=\sqrt{x+1}$. (See Example 8 )b. From the graph of $f$, is $f$ a one-to-one function?c. Write the domain of $f$ in interval notation.d. Write the range of $f$ in interval notation.e. Write an equation for $f^{1}(x)$.$\mathbf{f}$. Explain why the restriction $x \geq 0$ is placed on $f^{-1}$.g. Graph $y=f(x)$ and $y=f^{-1}(x)$ on the same coordinate system.h. Write the domain of $f^{-1}$ in interval notation.i. Write the range of $f^{-1}$ in interval notation.
a. Graph $f(x)=\sqrt{x-2}$.b. From the graph of $f$, is $f$ a one-to-one function?c. Write the domain of $f$ in interval notation.d. Write the range of $f$ in interval notation.e. Write an equation for $f^{1}(x)$.f. Explain why the restriction $x \geq 0$ is placed on $f^{-1}$.g. Graph $y=f(x)$ and $y=f^{-1}(x)$ on the same coordinate system.h. Write the domain of $f^{-1}$ in interval notation.i. Write the range of $f^{-1}$ in interval notation.
Given that the domain of a one-to-one function $f$ is $[0, \infty)$ and the range of $f$ is $[0,4),$ state the domain and range of $f^{-1}$.
Given that the domain of a one-to-one function $f$ is [-3,5) and the range of $f$ is $(-2, \infty),$ state the domain and range of $f^{-1}$.
Given $f(x)=|x|+3 ; x \leq 0,$ write an equation for $f^{-1}$
Given $f(x)=|x|-3 ; x \geq 0,$ write an equation for $f^{-1}$
If function $f$ adds 6 to $x$, then $f^{-1}$ __________ 6 from $x$. Function $f$ is defined by $f(x)=x+6$, and function $f^{-1}$ is defined by $f^{-1}(x)=$ _______________.
If function $f$ multiplies $x$ by $2,$ then $f^{-1}$ ____________ x by 2. Function $f$ is defined by $f(x)=2 x$, and function $f^{-1}$ is defined by $f^{-1}(x)=$ ___________.
Fill in the blanks and determine an equation for $f^{-1}(x)$ mentally.Suppose that function $f$ multiplies $x$ by 7 and subtracts4. Write an equation for $f^{-1}(x)$.
Fill in the blanks and determine an equation for $f^{-1}(x)$ mentally.Suppose that function $f$ divides $x$ by 3 and adds 11 . Write an equation for $f^{-1}(x)$.
Fill in the blanks and determine an equation for $f^{-1}(x)$ mentally.Suppose that function $f$ cubes $x$ and adds 20 . Write an equation for $f^{-1}(x)$.
Fill in the blanks and determine an equation for $f^{-1}(x)$ mentally.Suppose that function $f$ takes the cube root of $x$ and subtracts $10 .$ Write an equation for $f^{-1}(x)$.
Find the inverse mentally.. $f(x)=8 x+1$
Find the inverse mentally. $p(x)=2 x-10$
Find the inverse mentally. $q(x)=\sqrt[5]{x-4}+1$
Find the inverse mentally.$m(x)=\sqrt[3]{4 x}+3$
The graph of a function is given. Graph the inverse function.
The table defines $\mathbf{Y}_{1}=f(x)$ as a one-to-one function of $x .$ Find the values of $f^{-1}$ for the selected values of $x$.a. $f^{-1}(32)$b. $f^{-1}(-2.5)$c. $f^{-1}(26)$
The table defines $\mathbf{Y}_{1}=f(x)$ as a one-to-one function of $x .$ Find the values of $f^{-1}$ for the selected values of $x$.a. $f^{-1}(5)$b. $f^{-1}(9.45)$c. $f^{-1}(8)$
Determine if the statement is true or false.All linear functions with a nonzero slope have an inverse function.
Determine if the statement is true or false.The domain of any one-to-one function is the same as the domain of its inverse function.
Determine if the statement is true or false.The range of a one-to-one function is the same as the range of its inverse function.
Determine if the statement is true or false.No quadratic function defined by $f(x)=a x^{2}+b x+c$ $(a \neq 0)$ is one-to-one.
Suppose that during normal respiration, the volume of air inhaled per breath (called "tidal volume") by a mammal of any size is $6.33 \mathrm{~mL}$ per kilogram of body mass. a. Write a function representing the tidal volume $T(x)$ (in $\mathrm{mL}$ ) of a mammal of mass $x$ (in kg).b. Write an equation for $T^{-1}(x)$.c. What does the inverse function represent in the context of this problem?d. Find $T^{-1}(170)$ and interpret its meaning in context. Round to the nearest whole unit.
At a cruising altitude of $35,000 \mathrm{ft}$, a certain airplane travels $555 \mathrm{mph}$.a. Write a function representing the distance $d(t)$ (in mi) for $t$ hours at cruising altitude.b. Write an equation for $d^{-1}(t)$.c. What does the inverse function represent in the context of this problem?d. Evaluate $d^{-1}(2553)$ and interpret its meaning in context.
The millage rate is the amount of property tax per $\$ 1000$ of the taxable value of a home. For a certain county the millage rate is 24 mil. A city within the county also imposes a flat fee of $\$ 108$ per home.a. Write a function representing the total amount of property $\operatorname{tax} T(x)$ (in \$) for a home with a taxable value of $x$ thousand dollars.b. Write an equation for $T^{-1}(x)$.c. What does the inverse function represent in the context of this problem?d. Evaluate $T^{-1}(2988)$ and interpret its meaning in context.
Beginning on January 1 , park rangers in Everglades National Park began recording the water level for one particularly dry area of the park. The water level was initially $2.5 \mathrm{ft}$ and decreased by approximately $0.015 \mathrm{ft} /$ day. a. Write a function representing the water level $L(x)$ (in $\mathrm{ft}$ ), $x$ days after January $1 .$b. Write an equation for $L^{-1}(x)$.c. What does the inverse function represent in the context of this problem?d. Evaluate $L^{-1}(1.9)$ and interpret its meaning in context.
$V(r)=\frac{4}{3} \pi r^{3}$ gives the volume of a sphere as a function of its radius $r$. Find an equation for $r(V)$ and interpret its meaning in the context of this problem.
$F(C)=\frac{9}{5} C+32$ gives the temperature in degrees Fahrenheit as a function of the temperature $C$ in degrees Celsius. Find an equation for $C(F)$ and interpret its meaning in the context of this problem.
Explain the relationship between the domain and range of a one-to-one function $f$ and its inverse $f^{-1}$.
Write an informal definition of a one-to-one function.
Explain why if a horizontal line intersects the graph of a function in more than one point, then the function is not one-to-one.
Explain why the domain of $f(x)=x^{2}+k$ must be restricted to find an inverse function.
Consider a function defined as follows. Given $x,$ the value $f(x)$ is the exponent above the base of 2 that produces $x .$ For example, $f(16)=4$ because $2^{4}=16$ Evaluatea. $f(8)$b. $f(32)$c. $f(2)$d. $f\left(\frac{1}{8}\right)$
Consider a function defined as follows. Given $x,$ the value $f(x)$ is the exponent above the base of 3 that produces $x .$ For example, $f(9)=2$ because $3^{2}=9$ Evaluatea. $f(27)$b. $f(81)$c. $f(3)$d. $f\left(\frac{1}{9}\right)$
Show that every increasing function is one-to-one.
A function is said to be periodic if there exists some nonzero real number $p,$ called the period, such that $f(x+p)=f(x)$ for all real numbers $x$ in the domain of $f$. Explain why no periodic function is one-to-one.
Given the functions defined by $f(x)=2 x-1$ and $g(x)=\frac{x+1}{2}$, a. Graph $y=f(x), y=g(x),$ and the line $y=x .$ Does the graph suggest that $f$ and $g$ are inverses? Why?b. Enter the following functions into the graphing editor. ($$\mathrm{Y}_{1}=2 x-1$$$\mathrm{Y}_{2}=(x+1) / 2$$\mathrm{Y}_{3}=\mathrm{Y}_{1}\left(\mathrm{Y}_{2}\right)$$\mathrm{Y}_{4}=\mathrm{Y}_{2}\left(\mathrm{Y}_{1}\right)$c. Create a table of points showing $Y_{3}$ and $Y_{4}$ for several values of $x$. (Hint: Use the right and left arrows to scroll through the table editor to show functions $Y_{3}$ and $Y_{4}$.) Does the table suggest that $f$ and $g$ are inverses? Why?