# 1.7 Inverse functions  (Page 8/10)

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

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.

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

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

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

## 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.

how fast can i understand functions without much difficulty
what is set?
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this