Chapter 2, Problem 28RE

(a)

To determine

To find: the domain and the range of $f$ .

Answer to Problem 28RE

The domain of $f$ is $\left\{x|-5\le x\le 4\right\}or\left[-5,4\right]$ and range is $\left\{y|-3\le x\le 1\right\}or\left[-3,1\right]$

Explanation of Solution

Given information:

The graph of the function $f$ shown:

  

Calculation:

The domain of $f$ is $\left\{x|-5\le x\le 4\right\}or\left[-5,4\right]$ and range is $\left\{y|-3\le x\le 1\right\}or\left[-3,1\right]$

(b)

To determine

To find: the intervals on which $f$ is increasing, decreasing, or constant.

Answer to Problem 28RE

The function $f$ is increasing on $\left(-4,-2\right)$ and $\left(2,4\right)$ ; decreasing on $\left(-1,3\right)$ .

Explanation of Solution

Given information:

The graph of the function $f$ shown:

  

Calculation:

From graph, the function $f$ is increasing on $\left(-4,-2\right)$ and $\left(2,4\right)$ ; decreasing on $\left(-2,2\right)$ .

(c)

To determine

To find: local minimum values and local maximum values.

Answer to Problem 28RE

The local minimum values at $x=2$ and the local maximum value at $x=-2$ .

Explanation of Solution

Given information:

The graph of the function $f$ shown:

  

Calculation:

The local minimum values at $x=2$ and the local maximum value at $x=-2$ .

(d)

To determine

To find: the absolute maximum and absolute minimum value.

Answer to Problem 28RE

The absolute maximum is $f\left(-2\right)=1$ and absolute minimum at $f\left(2\right)=-1$ .

Explanation of Solution

Given information:

The graph of the function $f$ shown:

  

Calculation:

Absolute Maximum occurs at $\left(-2,1\right)$ .

Absolute Minimum occurs at $\left(2,-1\right)$ .

The absolute maximum is $f\left(-2\right)=1$ and absolute minimum at $f\left(2\right)=-1$ .

(e)

To determine

To find: whether the graph is symmetric with respect to the $x$ axis, the $y$ axis, or the origin.

Answer to Problem 28RE

The graph is symmetric about origin.

Explanation of Solution

Given information:

The graph of the function $f$ shown:

  

Calculation:

The symmetrical points on the graph are

  $\begin{array}{l}\left(4,3\right)\to \left(-4,-3\right)\\ \left(-2,1\right)\to \left(2,-1\right)\\ \left(3,0\right)\to \left(-3,0\right)\end{array}$

So, the graph is symmetric about origin.

(f)

To determine

To find: whether the function is even, odd, or neither.

Answer to Problem 28RE

The function is odd.

Explanation of Solution

Given information:

The graph of the function $f$ shown:

  

Calculation:

The graph is symmetric about origin.

So, the function is odd.

(g)

To determine

To find: the intercepts.

Answer to Problem 28RE

  $x$ intercepts: $-2,0,3$ and $y$ intercept: 0.

Explanation of Solution

Given information:

The graph of the function $f$ shown:

  

Calculation:

  $x$ intercepts are the points where the graph intersects the $x$ -axis.

  $y$ intercepts are the points where the graph intersects the $y$ -axis.

From graph, $x$ intercepts: $-2,0,3$ and $y$ intercept: 0.