Chapter 2, Problem 16RE

Consider the graph of the function $f$ on the right.

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

(b) List the intercepts.

(c) Find $f\left(-2\right)$.

(d) For what value of $x$ does $f\left(x\right)=-3$?

(e) Solve $f\left(x\right)>0$.

(f) Graph $y=f\left(x-3\right)$.

(g) Graph $y=f\left(\frac{1}{2}x\right)$.

(h) Graph $y=-f\left(x\right)$.

To determine

To find:

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

Answer to Problem 16RE

Solution:

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

Explanation of Solution

Given:

Calculation:

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

$\text{Hence domain }=\left[-4,3\right]$.

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

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

To determine

To find:

b. List the intercepts.

Answer to Problem 16RE

Solution:

b. $x$ and $y\text{-intercepts}$ are $\left(0,0\right)$.

Explanation of Solution

Given:

Calculation:

b. $x\text{-Intercept }=\left(0,0\right)$ and $y\text{-intercept }=\left(0,0\right)$.

To determine

To find:

c. $f\left(-2\right)$

Answer to Problem 16RE

Solution:

c. $-1$

Explanation of Solution

Given:

Calculation:

c. $f\left(-2\right)=-1\left(\text{as per the point in the graph}\right)$

To determine

To find:

d. For what value of $x$ does $f\left(x\right)=-3$?

Answer to Problem 16RE

Solution:

d. $-4$

Explanation of Solution

Given:

Calculation:

d. If $f\left(x\right)=-3$, then $x=-4$ (as per the point in the graph).

To determine

To find:

e. $x$ when $f\left(x\right)>0$.

Answer to Problem 16RE

Solution:

e. $\left(0,3\right]$

Explanation of Solution

Given:

Calculation:

e. When $f\left(x\right)>0$, then $x>0\text{ and }x\le 3$ in other words, $xϵ\left(0,3\right]$.

To determine

To find:

f. Graph $y=f\left(x–3\right)$.

Answer to Problem 16RE

Solution:

f. Graph of $y=f(x–3)$ is plotted.

Explanation of Solution

Given:

Calculation:

f. To get the graph $y=f\left(x-3\right)$, we shift the graph 3 units to the right.

The red dotted line shows the graph of $y=f\left(x-3\right)$.

To determine

To find:

g. Graph $y=f \left(\frac{1}{2}x\right)$.

Answer to Problem 16RE

Solution:

g. Graph of $y=f \left(\frac{1}{2}x\right)$ is plotted.

Explanation of Solution

Given:

Calculation:

g. To graph $y=f\left(\frac{1}{2}x\right)$, we stretch the graph of $y=f\left(x\right)$ by a factor of 2.

The red dotted line shows the graph of $y=f(\frac{x}{2})$.

To determine

To find:

h. Graph $y=-f(x)$.

Answer to Problem 16RE

Solution:

h. Graph of $y=-f\left(x\right)$ is plotted.

Explanation of Solution

Given:

Calculation:

h. To Graph $y=-f\left(x\right)$, we reflect the graph of $y=f\left(x\right)$.

The red dotted line shows the graph of $y=-f\left(x\right)$.