# 1.7 Inverse functions  (Page 7/10)

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For the following exercises, use function composition to verify that $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)\text{\hspace{0.17em}}$ are inverse functions.

$f\left(x\right)=\sqrt[3]{x-1}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)={x}^{3}+1$

$f\left(x\right)=-3x+5\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)=\frac{x-5}{-3}$

## Graphical

For the following exercises, use a graphing utility to determine whether each function is one-to-one.

$f\left(x\right)=\sqrt{x}$

one-to-one

$f\left(x\right)=\sqrt[3]{3x+1}$

$f\left(x\right)=-5x+1$

one-to-one

$f\left(x\right)={x}^{3}-27$

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 $\text{\hspace{0.17em}}f\left(x\right)=0.$

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

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 $\text{\hspace{0.17em}}f.$

## Numeric

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 $\text{\hspace{0.17em}}f\left(6\right)=7,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}\text{\hspace{0.17em}}{f}^{-1}\left(7\right).$

$6$

If $\text{\hspace{0.17em}}f\left(3\right)=2,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}{f}^{-1}\left(2\right).$

If $\text{\hspace{0.17em}}{f}^{-1}\left(-4\right)=-8,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}f\left(-8\right).$

$-4$

If $\text{\hspace{0.17em}}{f}^{-1}\left(-2\right)=-1,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}f\left(-1\right).$

For the following exercises, use the values listed in [link] to evaluate or solve.

 $x$ $f\left(x\right)$ 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 $\text{\hspace{0.17em}}f\left(x\right)=3.$

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 $f\left(x\right)$ 1 4 7 12 16
 $x$ 1 4 7 12 16 ${f}^{-1}\left(x\right)$ 3 6 9 13 14

## Technology

For the following exercises, find the inverse function. Then, graph the function and its inverse.

$f\left(x\right)=\frac{3}{x-2}$

$f\left(x\right)={x}^{3}-1$

${f}^{-1}\left(x\right)={\left(1+x\right)}^{1/3}$

Find the inverse function of $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{x-1}.\text{\hspace{0.17em}}$ Use a graphing utility to find its domain and range. Write the domain and range in interval notation.

## Real-world applications

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 $\text{\hspace{0.17em}}f\left(x\right)=\frac{9}{5}x+32.\text{\hspace{0.17em}}$ Find the inverse function, if it exists, and explain its meaning.

${f}^{-1}\left(x\right)=\frac{5}{9}\left(x-32\right).\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}C\left(r\right)=2\pi r.\text{\hspace{0.17em}}$ Express the radius of a circle as a function of its circumference. Call this function $\text{\hspace{0.17em}}r\left(C\right).\text{\hspace{0.17em}}$ Find $\text{\hspace{0.17em}}r\left(36\pi \right)\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}d\left(t\right)=50t.\text{\hspace{0.17em}}$ Find the inverse function by expressing the time of travel in terms of the distance traveled. Call this function $\text{\hspace{0.17em}}t\left(d\right).\text{\hspace{0.17em}}$ Find $\text{\hspace{0.17em}}t\left(180\right)\text{\hspace{0.17em}}$ and interpret its meaning.

$t\left(d\right)=\frac{d}{50},\text{\hspace{0.17em}}$ $t\left(180\right)=\frac{180}{50}.\text{\hspace{0.17em}}$ The time for the car to travel 180 miles is 3.6 hours.

## Functions and Function Notation

For the following exercises, determine whether the relation is a function.

$\left\{\left(a,b\right),\left(c,d\right),\left(e,d\right)\right\}$

function

$\left\{\left(5,2\right),\left(6,1\right),\left(6,2\right),\left(4,8\right)\right\}$

${y}^{2}+4=x,\text{\hspace{0.17em}}$ 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: $\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(-3\right);\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(2\right);\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(-a\right);\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-f\left(a\right);\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(a+h\right).$

$f\left(x\right)=-2{x}^{2}+3x$

$f\left(-3\right)=-27;$ $f\left(2\right)=-2;$ $f\left(-a\right)=-2{a}^{2}-3a;$
$-f\left(a\right)=2{a}^{2}-3a;$ $f\left(a+h\right)=-2{a}^{2}+3a-4ah+3h-2{h}^{2}$

$f\left(x\right)=2|3x-1|$

For the following exercises, determine whether the functions are one-to-one.

#### Questions & Answers

the sum of any two linear polynomial is what
divide simplify each answer 3/2÷5/4
divide simplify each answer 25/3÷5/12
Momo
how can are find the domain and range of a relations
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
can I see the picture
How would you find if a radical function is one to one?
how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
what is foci?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
i want to sure my answer of the exercise
what is the diameter of(x-2)²+(y-3)²=25
how to solve the Identity ?
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard