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For the following exercises, use function composition to verify that $\text{\hspace{0.17em}}f(x)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g(x)\text{\hspace{0.17em}}$ are inverse functions.
$f(x)=\sqrt[3]{x-1}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g(x)={x}^{3}+1$
$f(g(x))=x,\text{\hspace{0.17em}}g(f(x))=x$
$f(x)=-3x+5\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g(x)=\frac{x-5}{-3}$
For the following exercises, use a graphing utility to determine whether each function is one-to-one.
$f(x)=\sqrt{x}$
one-to-one
$f(x)=\sqrt[3]{3x+1}$
$f(x)=\mathrm{-5}x+1$
one-to-one
$f(x)={x}^{3}-27$
For the following exercises, determine whether the graph represents a one-to-one function.
not one-to-one
For the following exercises, use the graph of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ shown in [link] .
Find $\text{\hspace{0.17em}}f\left(0\right).$
$3$
Solve $\text{\hspace{0.17em}}f(x)=0.$
Find $\text{\hspace{0.17em}}{f}^{-1}\left(0\right).$
$2$
Solve $\text{\hspace{0.17em}}{f}^{-1}\left(x\right)=0.$
For the following exercises, use the graph of the one-to-one function shown in [link] .
Sketch the graph of $\text{\hspace{0.17em}}{f}^{-1}.\text{\hspace{0.17em}}$
Find $\text{\hspace{0.17em}}f(6)\text{and}{f}^{-1}(2).$
If the complete graph of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is shown, find the domain of $\text{\hspace{0.17em}}f.\text{\hspace{0.17em}}$
$\left[2,10\right]$
If the complete graph of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is shown, find the range of $\text{\hspace{0.17em}}f.$
For the following exercises, evaluate or solve, assuming that the function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is one-to-one.
If $\text{\hspace{0.17em}}f(6)=7,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}\text{\hspace{0.17em}}{f}^{-1}(7).$
$6$
If $\text{\hspace{0.17em}}f(3)=2,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}{f}^{-1}(2).$
If $\text{\hspace{0.17em}}{f}^{-1}\left(-4\right)=-8,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}f(-8).$
$-4$
If $\text{\hspace{0.17em}}{f}^{-1}\left(-2\right)=-1,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}f(-1).$
For the following exercises, use the values listed in [link] to evaluate or solve.
$x$ | $f(x)$ |
0 | 8 |
1 | 0 |
2 | 7 |
3 | 4 |
4 | 2 |
5 | 6 |
6 | 5 |
7 | 3 |
8 | 9 |
9 | 1 |
Find $\text{\hspace{0.17em}}f\left(1\right).$
$0$
Solve $\text{\hspace{0.17em}}f(x)=3.$
Find $\text{\hspace{0.17em}}{f}^{-1}\left(0\right).$
$\text{\hspace{0.17em}}1\text{\hspace{0.17em}}$
Solve $\text{\hspace{0.17em}}{f}^{-1}\left(x\right)=7.$
Use the tabular representation of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ in [link] to create a table for $\text{\hspace{0.17em}}{f}^{-1}\left(x\right).$
$x$ | 3 | 6 | 9 | 13 | 14 |
$f(x)$ | 1 | 4 | 7 | 12 | 16 |
$x$ | 1 | 4 | 7 | 12 | 16 |
${f}^{-1}(x)$ | 3 | 6 | 9 | 13 | 14 |
For the following exercises, find the inverse function. Then, graph the function and its inverse.
$f(x)=\frac{3}{x-2}$
$f(x)={x}^{3}-1$
${f}^{-1}(x)={(1+x)}^{1/3}$
Find the inverse function of $\text{\hspace{0.17em}}f(x)=\frac{1}{x-1}.\text{\hspace{0.17em}}$ Use a graphing utility to find its domain and range. Write the domain and range in interval notation.
To convert from $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ degrees Celsius to $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ degrees Fahrenheit, we use the formula $\text{\hspace{0.17em}}f(x)=\frac{9}{5}x+32.\text{\hspace{0.17em}}$ Find the inverse function, if it exists, and explain its meaning.
${f}^{-1}(x)=\frac{5}{9}\left(x-32\right).\text{\hspace{0.17em}}$ Given the Fahrenheit temperature, $\text{\hspace{0.17em}}x,\text{\hspace{0.17em}}$ this formula allows you to calculate the Celsius temperature.
The circumference $\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ of a circle is a function of its radius given by $\text{\hspace{0.17em}}C(r)=2\pi r.\text{\hspace{0.17em}}$ Express the radius of a circle as a function of its circumference. Call this function $\text{\hspace{0.17em}}r(C).\text{\hspace{0.17em}}$ Find $\text{\hspace{0.17em}}r(36\pi )\text{\hspace{0.17em}}$ and interpret its meaning.
A car travels at a constant speed of 50 miles per hour. The distance the car travels in miles is a function of time, $\text{\hspace{0.17em}}t,\text{\hspace{0.17em}}$ in hours given by $\text{\hspace{0.17em}}d(t)=50t.\text{\hspace{0.17em}}$ Find the inverse function by expressing the time of travel in terms of the distance traveled. Call this function $\text{\hspace{0.17em}}t(d).\text{\hspace{0.17em}}$ Find $\text{\hspace{0.17em}}t(180)\text{\hspace{0.17em}}$ and interpret its meaning.
$t(d)=\frac{d}{50},\text{\hspace{0.17em}}$ $t(180)=\frac{180}{50}.\text{\hspace{0.17em}}$ The time for the car to travel 180 miles is 3.6 hours.
For the following exercises, determine whether the relation is a function.
$\left\{(a,b),(c,d),(e,d)\right\}$
function
$\left\{(5,2),(6,1),(6,2),(4,8)\right\}$
${y}^{2}+4=x,\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ the independent variable and $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ the dependent variable
not a function
Is the graph in [link] a function?
For the following exercises, evaluate the function at the indicated values: $\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}f(-3);\text{\hspace{0.17em}}\text{\hspace{0.17em}}f(2);\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}f(-a);\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-f(a);\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}f(a+h).$
$f(x)=-2{x}^{2}+3x$
$f(-3)=-27;$
$f(2)=-2;$
$f(-a)=-2{a}^{2}-3a;$
$-f(a)=2{a}^{2}-3a;$
$f(a+h)=-2{a}^{2}+3a-4ah+3h-2{h}^{2}$
$f(x)=2\left|3x-1\right|$
For the following exercises, determine whether the functions are one-to-one.
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