Chapter 2, Problem 17RE

Use the graph of the function $f$ shown to find:

(a) The domain and the range of $f$.

(b) The intervals on which $f$ is increasing, decreasing, or constant.

(c) The local minimum values and local maximum values.

(d) The absolute maximum and absolute minimum.

(e) Whether the graph is symmetric with respect to the $x\text{-axis}$, the $y\text{-axis}$, or the origin.

(f) Whether the function is even, odd, or neither.

(g) The intercepts, if any.

To determine

To find:

a. The domain and the range of $f$.

Answer to Problem 17RE

Solution:

a. $\text{Domain}:\left[-4,4\right];\text{ Range:}\left[-3,3\right]$.

Explanation of Solution

Given:

Calculation:

a. Domain: set of all the values of $x$ for which graph is defined.

Hence domain $=\left[-4,4\right]$.

Range: set of all the values of $y$ for which graph is defined.

Hence Range $=\left[-3,3\right]$.

To determine

To find:

b. The intervals on which $f$ is increasing, decreasing, or constant.

Answer to Problem 17RE

Solution:

b. The function is increasing in the interval $\left(\infty ,-2\right)\cup \left(2,4\right)$.

The function is decreasing in the interval $(2,2)$.

The function is not constant on any interval.

Explanation of Solution

Given:

Calculation:

b. From the graph we see that, the function is increasing in the interval $(-\infty ,-2)$ and $\left(2,4\right)$ that is $\left(\infty ,-2\right)\cup \left(2,4\right)$.

The function is decreasing in the interval $(-2,2)$.

The function is not constant on any interval.

To determine

To find:

c. The local minimum values and local maximum values.

Answer to Problem 17RE

Solution:

c. local maximum value is $f\left(-2\right)=1$ and $f\left(2\right)=-1$. There is no Local minimum.

Explanation of Solution

Given:

Calculation:

c. Local maximum exists at $x=-2$ and the local maximum value is $f\left(-2\right)=1$.

Local maximum exists at $x=2$ and the local maximum value is $f\left(2\right)=-1$.

The graph does not have local minimum.

To determine

To find:

d. The absolute maximum and absolute minimum.

Answer to Problem 17RE

Solution:

d. The graph has absolute maximum at $\left(4,3\right)$ and it does not have absolute minimum.

Explanation of Solution

Given:

Calculation:

d. The graph has absolute maximum at $\left(4,3\right)$ and it does not have absolute minimum.

To determine

To find:

e. Whether the graph is symmetric with respect to the $x\text{-axis}$, the $y\text{-axis}$, or the origin.

Answer to Problem 17RE

Solution:

e. The Graph is not symmetric with respect to the $x\text{-axis}$, the $y\text{-axis}$, or the origin.

Explanation of Solution

Given:

Calculation:

e. The Graph is not symmetric with respect to the $x\text{-axis}$, the $y\text{-axis}$, or the origin.

To determine

To find:

f. Whether the function is even, odd, or neither.

Answer to Problem 17RE

Solution:

f. The function is neither even nor odd.

Explanation of Solution

Given:

Calculation:

f. From $\left(e\right)$, we see that the graph is not symmetric with respect to the $x\text{-axis}$, the $y\text{-axis}$ or origin. Therefore, the function is neither even nor odd.

To determine

To find:

g. The intercepts, if any.

Answer to Problem 17RE

Solution:

g. $x\text{-intercept }-\left(-3,0\right),\left(0,0\right),\left(3,0\right)$.

$y\text{-intercept }-\left(0,0\right)$.

Explanation of Solution

Given:

Calculation:

g. $x\text{-intercepts}$ are those where the graph meets $x\text{-axis}$ or has $y\text{-coordinate}$ 0, $x\text{-intercept }-\left(-3,0\right),\left(0,0\right),\left(3,0\right)$.

$y\text{-intercepts}$ are those where the graph meets $y\text{-axis}$ or has $x\text{-coordinate}$ 0,$y\text{-intercept }-\left(0,0\right)$.