# 3.7 Inverse functions  (Page 8/9)

 Page 8 / 9

$f\left(x\right)=-3x+5$

one-to-one

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

For the following exercises, use the vertical line test to determine if the relation whose graph is provided is a function.

function

function

For the following exercises, graph the functions.

$f\left(x\right)=|x+1|$

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

For the following exercises, use [link] to approximate the values.

$f\left(2\right)$

$f\left(-2\right)$

$2$

If $\text{\hspace{0.17em}}f\left(x\right)=-2,\text{\hspace{0.17em}}$ then solve for $\text{\hspace{0.17em}}x.$

If $\text{\hspace{0.17em}}f\left(x\right)=1,\text{\hspace{0.17em}}$ then solve for $\text{\hspace{0.17em}}x.$

or

For the following exercises, use the function $\text{\hspace{0.17em}}h\left(t\right)=-16{t}^{2}+80t\text{\hspace{0.17em}}$ to find the values in simplest form.

$\frac{h\left(2\right)-h\left(1\right)}{2-1}$

$\frac{h\left(a\right)-h\left(1\right)}{a-1}$

$\frac{-64+80a-16{a}^{2}}{-1+a}=-16a+64$

## Domain and Range

For the following exercises, find the domain of each function, expressing answers using interval notation.

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

$f\left(x\right)=\frac{x-3}{{x}^{2}-4x-12}$

$\left(-\infty ,-2\right)\cup \left(-2,6\right)\cup \left(6,\infty \right)$

$f\left(x\right)=\frac{\sqrt{x-6}}{\sqrt{x-4}}$

Graph this piecewise function:

## Rates of Change and Behavior of Graphs

For the following exercises, find the average rate of change of the functions from

$f\left(x\right)=4x-3$

$f\left(x\right)=10{x}^{2}+x$

$31$

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

For the following exercises, use the graphs to determine the intervals on which the functions are increasing, decreasing, or constant.

increasing $\text{\hspace{0.17em}}\left(2,\infty \right);\text{\hspace{0.17em}}$ decreasing $\text{\hspace{0.17em}}\left(-\infty ,2\right)$

increasing $\text{}\left(-3,1\right);\text{}$ constant $\text{\hspace{0.17em}}\left(-\infty ,-3\right)\cup \left(1,\infty \right)$

Find the local minimum of the function graphed in [link] .

Find the local extrema for the function graphed in [link] .

local minimum $\text{\hspace{0.17em}}\left(-2,-3\right);\text{\hspace{0.17em}}$ local maximum $\text{\hspace{0.17em}}\left(1,3\right)$

For the graph in [link] , the domain of the function is $\text{\hspace{0.17em}}\left[-3,3\right].$ The range is $\text{\hspace{0.17em}}\left[-10,10\right].\text{\hspace{0.17em}}$ Find the absolute minimum of the function on this interval.

Find the absolute maximum of the function graphed in [link] .

$\text{\hspace{0.17em}}\left(-1.8,10\right)\text{\hspace{0.17em}}$

## Composition of Functions

For the following exercises, find $\text{\hspace{0.17em}}\left(f\circ g\right)\left(x\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(g\circ f\right)\left(x\right)\text{\hspace{0.17em}}$ for each pair of functions.

$f\left(x\right)=4-x,\text{\hspace{0.17em}}g\left(x\right)=-4x$

$f\left(x\right)=3x+2,\text{\hspace{0.17em}}g\left(x\right)=5-6x$

$\left(f\circ g\right)\left(x\right)=17-18x;\text{\hspace{0.17em}}\left(g\circ f\right)\left(x\right)=-7-18x$

$f\left(x\right)={x}^{2}+2x,\text{\hspace{0.17em}}g\left(x\right)=5x+1$

$\left(f\circ g\right)\left(x\right)=\sqrt{\frac{1}{x}+2};\text{\hspace{0.17em}}\left(g\circ f\right)\left(x\right)=\frac{1}{\sqrt{x+2}}$

For the following exercises, find $\text{\hspace{0.17em}}\left(f\circ g\right)\text{\hspace{0.17em}}$ and the domain for $\text{\hspace{0.17em}}\left(f\circ g\right)\left(x\right)\text{\hspace{0.17em}}$ for each pair of functions.

$\left(f\circ g\right)\left(x\right)=\frac{1}{\sqrt{x}},\text{\hspace{0.17em}}x>0$

For the following exercises, express each function $\text{\hspace{0.17em}}H\text{\hspace{0.17em}}$ as a composition of two functions $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}H\left(x\right)=\left(f\circ g\right)\left(x\right).$

$H\left(x\right)=\sqrt{\frac{2x-1}{3x+4}}$

sample: $\text{\hspace{0.17em}}g\left(x\right)=\frac{2x-1}{3x+4};\text{\hspace{0.17em}}f\left(x\right)=\sqrt{x}$

$H\left(x\right)=\frac{1}{{\left(3{x}^{2}-4\right)}^{-3}}$

## Transformation of Functions

For the following exercises, sketch a graph of the given function.

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

$f\left(x\right)={\left(x+4\right)}^{3}$

$f\left(x\right)=\sqrt{x}+5$

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

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

$f\left(x\right)=5\sqrt{-x}-4$

$f\left(x\right)=4\left[|x-2|-6\right]$

$f\left(x\right)=-{\left(x+2\right)}^{2}-1$

For the following exercises, sketch the graph of the function $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ if the graph of the function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is shown in [link] .

$g\left(x\right)=f\left(x-1\right)$

$g\left(x\right)=3f\left(x\right)$

For the following exercises, write the equation for the standard function represented by each of the graphs below.

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

For the following exercises, determine whether each function below is even, odd, or neither.

$f\left(x\right)=3{x}^{4}$

even

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

$h\left(x\right)=\frac{1}{x}+3x$

odd

For the following exercises, analyze the graph and determine whether the graphed function is even, odd, or neither.

even

## Absolute Value Functions

For the following exercises, write an equation for the transformation of $\text{\hspace{0.17em}}f\left(x\right)=|x|.$

$f\left(x\right)=\frac{1}{2}|x+2|+1$

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

For the following exercises, graph the absolute value function.

$f\left(x\right)=|x-5|$

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

$f\left(x\right)=|2x-4|$

## Inverse Functions

For the following exercises, find for each function.

$f\left(x\right)=9+10x$

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

${f}^{-1}\left(x\right)=\frac{-2x}{x-1}$

For the following exercise, find a domain on which the function is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of restricted to that domain.

$f\left(x\right)={x}^{2}+1$

Given $f\left(x\right)={x}^{3}-5$ and $g\left(x\right)=\sqrt[3]{x+5}:$

1. Find and $g\left(f\left(x\right)\right).$
2. What does the answer tell us about the relationship between $f\left(x\right)$ and $g\left(x\right)?$
1. and $g\left(f\left(x\right)\right)=x.$
2. This tells us that $f$ and $g$ are inverse functions

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

$f\left(x\right)=\frac{1}{x}$

The function is one-to-one.

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

The function is not one-to-one.

If $f\left(5\right)=2,$ find ${f}^{-1}\left(2\right).$

$5$

If $f\left(1\right)=4,$ find ${f}^{-1}\left(4\right).$

## Practice test

For the following exercises, determine whether each of the following relations is a function.

$y=2x+8$

The relation is a function.

$\left\{\left(2,1\right),\left(3,2\right),\left(-1,1\right),\left(0,-2\right)\right\}$

For the following exercises, evaluate the function $\text{\hspace{0.17em}}f\left(x\right)=-3{x}^{2}+2x\text{\hspace{0.17em}}$ at the given input.

$f\left(-2\right)$

−16

$\text{\hspace{0.17em}}f\left(a\right)\text{\hspace{0.17em}}$

Show that the function $\text{\hspace{0.17em}}f\left(x\right)=-2{\left(x-1\right)}^{2}+3\text{\hspace{0.17em}}$ is not one-to-one.

The graph is a parabola and the graph fails the horizontal line test.

Write the domain of the function $\text{\hspace{0.17em}}f\left(x\right)=\sqrt{3-x}\text{\hspace{0.17em}}$ in interval notation.

Given $\text{\hspace{0.17em}}f\left(x\right)=2{x}^{2}-5x,\text{\hspace{0.17em}}$ find $f\left(a+1\right)-f\left(1\right)\text{\hspace{0.17em}}$ in simplest form.

$2{a}^{2}-a$

Graph the function

Find the average rate of change of the function $\text{\hspace{0.17em}}f\left(x\right)=3-2{x}^{2}+x\text{\hspace{0.17em}}$ by finding $\text{\hspace{0.17em}}\frac{f\left(b\right)-f\left(a\right)}{b-a}\text{\hspace{0.17em}}$ in simplest form.

$-2\left(a+b\right)+1$

For the following exercises, use the functions to find the composite functions.

$\left(g\circ f\right)\left(x\right)$

$\left(g\circ f\right)\left(1\right)$

$\sqrt{2}$

Express $\text{\hspace{0.17em}}H\left(x\right)=\sqrt[3]{5{x}^{2}-3x}\text{\hspace{0.17em}}$ as a composition of two functions, $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g,\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}\left(f\circ g\right)\left(x\right)=H\left(x\right).$

For the following exercises, graph the functions by translating, stretching, and/or compressing a toolkit function.

$f\left(x\right)=\sqrt{x+6}-1$

$f\left(x\right)=\frac{1}{x+2}-1$

For the following exercises, determine whether the functions are even, odd, or neither.

$f\left(x\right)=-\frac{5}{{x}^{2}}+9{x}^{6}$

$\text{even}$

$f\left(x\right)=-\frac{5}{{x}^{3}}+9{x}^{5}$

$f\left(x\right)=\frac{1}{x}$

$\text{odd}$

Graph the absolute value function $\text{\hspace{0.17em}}f\left(x\right)=-2|x-1|+3.$

For the following exercises, find the inverse of the function.

$f\left(x\right)=3x-5$

${f}^{-1}\left(x\right)=\frac{x+5}{3}$

$f\left(x\right)=\frac{4}{x+7}$

For the following exercises, use the graph of $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ shown in [link] .

On what intervals is the function increasing?

On what intervals is the function decreasing?

Approximate the local minimum of the function. Express the answer as an ordered pair.

$\left(1.1,-0.9\right)$

Approximate the local maximum of the function. Express the answer as an ordered pair.

For the following exercises, use the graph of the piecewise function shown in [link] .

Find $\text{\hspace{0.17em}}f\left(2\right).$

$f\left(2\right)=2$

Find $\text{\hspace{0.17em}}f\left(-2\right).$

Write an equation for the piecewise function.

$f\left(x\right)=\left\{\begin{array}{c}|x|\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\le 2\\ 3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x>2\end{array}$

For the following exercises, use the values listed in [link] .

$x$ $F\left(x\right)$
0 1
1 3
2 5
3 7
4 9
5 11
6 13
7 15
8 17

Find $\text{\hspace{0.17em}}F\left(6\right).$

Solve the equation $\text{\hspace{0.17em}}F\left(x\right)=5.$

$x=2$

Is the graph increasing or decreasing on its domain?

Is the function represented by the graph one-to-one?

yes

Find $\text{\hspace{0.17em}}{F}^{-1}\left(15\right).$

Given $\text{\hspace{0.17em}}f\left(x\right)=-2x+11,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}{f}^{-1}\left(x\right).$

${f}^{-1}\left(x\right)=-\frac{x-11}{2}$

what is math number
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
what is the coefficient of -4×
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
An investment account was opened with an initial deposit of \$9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
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salma
Commplementary angles
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Sherica
im all ears I need to learn
Sherica
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Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar