Chapter 2, Problem 38RE

To determine

To answer:

a. The questions related to the function $f(x)=\{\begin{array}{l}x\text{if}-4\le x<0\\ 1\text{if }x=0\\ 3x\text{if}x>0\end{array}$.

Answer to Problem 38RE

Solution:

a. The domain of the function is $[-4,\infty )$.

Explanation of Solution

Given:

$f(x)=\{\begin{array}{l}x\text{if}-4\le x<0\\ 1\text{if }x=0\\ 3x\text{if}x>0\end{array}$

Calculation:

a. Domain of the function:

The domain of $f\left(x\right)=x$ is $-4\le x<0$.

The domain of $f\left(x\right)=1$ is $x=0$.

The domain of $f\left(x\right)=3x$ is $x>0$.

Combing both the domains, we get the domain all the pieces.

Thus, the domain of the piecewise function is $x>-4$ or $[-4,\infty )$

To determine

To answer:

b. The questions related to the function $f(x)=\{\begin{array}{l}x\text{if}-4\le x<0\\ 1\text{if }x=0\\ 3x\text{if}x>0\end{array}$.

Answer to Problem 38RE

Solution:

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

Explanation of Solution

Given:

$f(x)=\{\begin{array}{l}x\text{if}-4\le x<0\\ 1\text{if }x=0\\ 3x\text{if}x>0\end{array}$

Calculation:

b. Intercepts of the function:

The $y\text{-intercepts}$ of the function is $\left(0, 1\right)$.

To determine

To answer:

c. The questions related to the function $f(x)=\{\begin{array}{l}x\text{if}-4\le x<0\\ 1\text{if }x=0\\ 3x\text{if}x>0\end{array}$.

Answer to Problem 38RE

Solution:

c.

Explanation of Solution

Given:

$f(x)=\{\begin{array}{l}x\text{if}-4\le x<0\\ 1\text{if }x=0\\ 3x\text{if}x>0\end{array}$

Calculation:

c. Graph of the piecewise function:

The graph the piecewise function $f(x)=\{\begin{array}{l}x\text{if}-4\le x<0\\ 1\text{if }x=0\\ 3x\text{if}x>0\end{array}$ is shown below:

To determine

To answer:

d. The questions related to the function $f(x)=\{\begin{array}{l}x\text{if}-4\le x<0\\ 1\text{if }x=0\\ 3x\text{if}x>0\end{array}$.

Answer to Problem 38RE

Solution:

d. The range of the function is $[-4, 0)\cup \left(0, \infty \right)$.

Explanation of Solution

Given:

$f(x)=\{\begin{array}{l}x\text{if}-4\le x<0\\ 1\text{if }x=0\\ 3x\text{if}x>0\end{array}$

Calculation:

d. Range of the function:

From the graph, we find that the range is $[-4, 0)\cup \left(0, \infty \right)$.