# 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.

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
how solve standard form of polar
what is a complex number used for?
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
Is there any rule we can use to get the nth term ?
how do you get the (1.4427)^t in the carp problem?
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
hello
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.
if you have the amplitude and the period and the phase shift ho would you know where to start and where to end?
rotation by 80 of (x^2/9)-(y^2/16)=1
thanks the domain is good but a i would like to get some other examples of how to find the range of a function
what is the standard form if the focus is at (0,2) ?
a²=4