Chapter 2, Problem 25RE

(a)

To determine

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

Answer to Problem 25RE

The range of $f$ is $\left\{y|3\le x\le -3\right\}$ or the interval $\left[3,-3\right]$ .

Explanation of Solution

Given information:

The graph of the function $f$ shown:

  

Calculation:

The points on the graph of $f$ have $x$ coordinate between $-4$ and 3, inclusive. So, for given number $x$ between $-4$ and 3,there is a point $\left(x,f\left(x\right)\right)$ on the graph.

Therefore, the domain of $f$ is $\left\{x|-4\le x\le 3\right\}$ or the interval $\left[-4,3\right]$ .

The points on the graph of $f$ all have the $y$ coordinate between 3 and $-3$ , inclusive.

For given value if $y$ between 3 and $-3$ , there exists at least on number $x$ in the domain.

So, the range of $f$ is $\left\{y|3\le x\le -3\right\}$ or the interval $\left[3,-3\right]$ .

(b)

To determine

To list: the intercepts.

Answer to Problem 25RE

The origin $\left(0,0\right)$ is the only intercept.

Explanation of Solution

Given information:

The graph of the function $f$ shown:

  

Calculation:

Intercepts are the points where the graph touches the coordinate axes. The $x$ intercepts have the $y$ coordinate 0, and the $y$ intercepts have the $x$ coordinates 0.

From the graph that the origin $\left(0,0\right)$ is the only intercept.

(c)

To determine

To find: the value of $f\left(-2\right)$ .

Answer to Problem 25RE

  $f\left(-2\right)=-1$

Explanation of Solution

Given information:

The graph of the function $f$ shown:

  

Calculation:

  $f\left(-2\right)$ is the value of $y$ on the graph when the value of $x$ is $-2$ .

The point $\left(-2,-1\right)$ on the graph corresponds to this situation.

Thus, $f\left(-2\right)=-1$

(d)

To determine

To find: the value of $x$ when $f\left(x\right)=-3$ .

Answer to Problem 25RE

  $f\left(x\right)=-3$ , when $x=-4$ .

Explanation of Solution

Given information:

The graph of the function $f$ shown:

  

Calculation:

  $f\left(x\right)=-3$ means that the value of $y$ on the graph is $-3$ at a given $x$ value. Look for points on the graph for which the $y$ coordinate is $-3$ .

The point $\left(-4,-3\right)$ on the graph corresponds to this situation.

Thus, $f\left(x\right)=-3$ , when $x=-4$ .

(e)

To determine

To solve: $f\left(x\right)>0$

Answer to Problem 25RE

Using inequality notations, $f\left(x\right)>0$ for $0\le x\le 3$ .

Explanation of Solution

Given information:

The graph of the function $f$ shown:

  

Calculation:

The function $f\left(x\right)>0$ indicates that the $y$ coordinate should have positive values for given value of the $x$ coordinate.

In the graph, $x$ values from $-4$ to 3 are shown. Determine for which of these values the $y$ coordinate is positive.

The point occur on $\left[0,3\right]$ .

So, using inequality notations, $f\left(x\right)>0$ for $0\le x\le 3$ .

(f)

To determine

To sketch: the graph of $y=f\left(x-3\right)$

Explanation of Solution

Given information:

The graph of the function $f$ shown:

  

Calculation:

Since 3 is subtracted from $f\left(x\right)$ , the graph of $y=f\left(x-3\right)$ is obtained by shifting the graph of $f$ horizontally to the right by 3 units.

  

(g)

To determine

To sketch: the graph of $y=f\left(\frac{1}{2}x\right)$

Explanation of Solution

Given information:

The graph of the function $f$ shown:

  

Calculation:

The graph of $y=f\left(\frac{1}{2}x\right)$ is the horizontally compressed version of the graph of $f\left(x\right)$ by a factor of $\frac{1}{2}$ . The function $f\left(x\right)$ is graphed by multiplying given $x$ coordinate of $f\left(x\right)$ by $\frac{1}{2}$ .

  

(h)

To determine

To sketch: the graph of $y=-f\left(x\right)$

Explanation of Solution

Given information:

The graph of the function $f$ shown:

  

Calculation:

The graph of the function $-f\left(x\right)$ is the reflection of the graph of $f\left(x\right)$ about the $y$ axis. The function $f\left(x\right)$ is graphed by multiplying the $x$ coordinates of $f\left(x\right)$ by $-1$ .