# 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 $\text{\hspace{0.17em}}{f}^{-1}\left(x\right).$

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

#### Questions & Answers

what are you up to?
Mark Reply
nothing up todat yet
Miranda
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jai
hello
jai
Miranda Drice
jai
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jai
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Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
Propessor Reply
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
sita Reply
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
What is algebra
Pearl Reply
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
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Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
which grade are you in though
Miranda
oh woww I understand
Miranda
haha. already finished college
Jeffrey
how about you? what grade are you now?
Jeffrey
I'm going to 11grade
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
yes steve. you're right
Jeffrey
so you better
Miranda
what is the solution of the given equation?
Nelson Reply
which equation
Miranda
I dont know. lol
Jeffrey
please where is the equation
Miranda
ask nelson. lol
Jeffrey
answer and questions in exercise 11.2 sums
Yp Reply
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
Swadesh
cos(- z)=cos z
Mustafa
what is a algebra
Jallah Reply
(x+x)3=?
Narad
6x
Obed
what is the identity of 1-cos²5x equal to?
liyemaikhaya Reply
__john __05
Kishu
Hi
Abdel
hi
Ye
hi
Nokwanda
C'est comment
Abdel
Hi
Amanda
hello
SORIE
Hiiii
Chinni
hello
Ranjay
hi
ANSHU
hiiii
Chinni
h r u friends
Chinni
yes
Hassan
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SORIE
I speak French
Abdel
okay no problem since we gather here and get to know each other
SORIE
hi im stupid at math and just wanna join here
Yaona
lol nahhh none of us here are stupid it's just that we have Fast, Medium, and slow learner bro but we all going to work things out together
SORIE
it's 12
what is the function of sine with respect of cosine , graphically
Karl Reply
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
Aashish Reply
sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
I don't understand how radicals works pls
Kenny Reply
How look for the general solution of a trig function
collins Reply

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