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### Precalculus Enhanced with Graphing Utilities

Book edition 6th
Author(s) Sullivan
Pages 1200 pages
ISBN 9780321795465

# For the graph of the function f,(a) Find the domain and the range of f.(b) Find the intercepts.(c) Is the graph of f symmetric with respect to the x-axis, the y-axis, or the origin?(d) Find f (2).(e) For what value(s) of x is $f\left(x\right)=3$?(f) Solve $f\left(x\right)<0$.(g) Graph $y=f\left(x\right)+2$.(h) Graph $y=f\left(-x\right)$.(i) Graph $y=2f\left(x\right)$.(j) Is f even, odd, or neither?(k) Find the interval(s) on which f is increasing.

(a) The domain of f is ${D}_{f}=\left[-4,4\right]$ and the range is $R=\left[-1,3\right]$

(b) The intercepts are $\left(-1,0\right),\left(0,-1\right),\left(1,0\right)$

(c) The x coordinates of the corresponding pairs are symmetric with respect to the y-axis, so the f is symmetric with respect to the y-axis.

(d) $f\left(2\right)=1$

(e) The values of x are $\left(4,3\right)\to f\left(4\right)=3\phantom{\rule{0ex}{0ex}}\left(-4,3\right)\to f\left(-4\right)=3$

(f) localid="1650165486015" $x\in \left[-1,1\right]$

(g) The graph is

(h) The graph is

(i) The graph is

(j) The function f is even

(k) The function f is increasing $x\in \left[0,4\right]$

See the step by step solution

## Step 1. Given information

For the graph of the function f.

## Part (a) of Step 1. Domain and Range

The domain is a set of the numbers x for which the function is defined. ${D}_{f}=\left[-4,4\right]$

The range is a set of all the values of the function, ${R}_{f}=\left[0,3\right]$

## Part (b) of Step 1. Intercept

To find the intercept, check where the graph of the function f intersects the x-axis and the y--axis.

The intercepts are $\left(-1,0\right),\left(0,-1\right),\left(1,0\right)$

## Part (c) of Step 1. Is the graph symmetric

To find if the graph of f is symmetric with respect to the x-axis, the y-axis, or the origin check the different values of x, the function has the same value,

$\left(-4,3\right),\left(4,3\right)\phantom{\rule{0ex}{0ex}}\left(-2,1\right),\left(2,1\right)\phantom{\rule{0ex}{0ex}}\left(-1,0\right),\left(1,0\right)$

We can notice that the x- coordinate of the corresponding pairs is symmetric with respect to the y-axis, so the f is symmetric with respect to the y-axis.

## Part (d) of Step 1. find f2

To find the value of $f\left(2\right)$ find the value of the function if $x=2$

$f\left(2\right)=1$

## Part (e) of Step 1. Value of x if fx=3

As per the graph, we can notice that the solution are:

$\left(4,3\right)\to f\left(4\right)=3\phantom{\rule{0ex}{0ex}}\left(-4,3\right)\to f\left(-4\right)=3$

## Part (f) of Step 1. Solve

To find the values of x when the values of the function are less than zero, we have to find where is the graph of the function f under x-axis.

As per the graph $x\in \left[-1,1\right]$

## Part (g) of Step 1. Graph

If a constant k is added to the function, then there will be a vertical shift in the graph that is $k>0$.

If $k<0$ then the graph will move k unit downwards.

As per the question, we add 2 to the given function, therefore the graph will be

## Part (h) of Step 1. Graph

To graph $y=f\left(-x\right)$, we must reflect the graph of f along the y-axis.

The graph of f is symmetrical to the y-axis, therefore, the graph of $y=f\left(-x\right)$ should be the same with f

## Part (i) of Step 1.  Graph

If a function is multiplied by a scalar factor k, then it will result to a vertical stretch to the graph of the function

if $k>1$, the graph is stretched vertically or narrow

If $0 the graph is compressed or widens.

If, $k<0$, the graph is stretched and reflected along the x-axis

if the function is multiplied by positive 2, therefore the graph narrows or stretched vertically

The graph of $y=2f\left(x\right)$ is

## Part (j) of Step 1. Is f even, odd, or neither

The even functions are symmetric with respect to the y-axis and the odd functions are symmetric with respect to the origin. As per part (c) we have the graph of the f, is symmetric over the y-axis, so the function f is even.

## Part (k) of Step 1. interval(s) on winch f is increasing

If we increase the value of the argument x, the function values rise. So we have to find those z values.

As per the graph, the f is increasing:

$x\in \left[0,4\right]$