Chapter 2, Problem 37RE

In Problems 37 and 38:

(a) Find the domain of each function.

(b) Locate any intercepts.

(c) Graph each function.

(d) Based on the graph, find the range.

$f\left(x\right)\text{ = }\{\begin{array}{l}3xif-\text{ 2 < }x\le \text{ 1}\\ x\text{ + 1 }ifx\text{ > 1}\end{array}$

To determine

To answer:

a. The questions related to the function $f\left(x\right)=\{\begin{array}{l}3x\text{}\text{if}-2<x\le 1\\ x+1\text{if}x>1\end{array}$.

Answer to Problem 37RE

Solution:

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

Explanation of Solution

Given:

$f\left(x\right)=\{\begin{array}{l}3x\text{}\text{if}-2<x\le 1\\ x+1\text{if}x>1\end{array}$

Calculation:

a. Domain of the function:

The domain of $f\left(x\right)=3x$ is $-2<x\le 1$.

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

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

Thus, the domain of the piecewise function is $x>-2$ or $\left(-2,\infty \right)$.

To determine

To answer:

b. The questions related to the function $f\left(x\right)=\{\begin{array}{l}3x\text{}\text{if}-2<x\le 1\\ x+1\text{if}x>1\end{array}$.

Answer to Problem 37RE

Solution:

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

Explanation of Solution

Given:

$f\left(x\right)=\{\begin{array}{l}3x\text{}\text{if}-2<x\le 1\\ x+1\text{if}x>1\end{array}$

Calculation:

b. Intercepts of the function:

The $x$ and $y$ intercepts of the function $f\left(x\right)=3x$ are $\left(0, 0\right)$.

To determine

To answer:

c. The questions related to the function $f\left(x\right)=\{\begin{array}{l}3x\text{}\text{if}-2<x\le 1\\ x+1\text{if}x>1\end{array}$.

Answer to Problem 37RE

Solution:

c.

Explanation of Solution

Given:

$f\left(x\right)=\{\begin{array}{l}3x\text{}\text{if}-2<x\le 1\\ x+1\text{if}x>1\end{array}$

Calculation:

c. Graph of the piecewise function:

The graph the piecewise function $f\left(x\right)=\{\begin{array}{l}3x\text{}\text{if}-2<x\le 1\\ x+1\text{if}x>1\end{array}$ is shown below:

To determine

To answer:

d. The questions related to the function $f\left(x\right)=\{\begin{array}{l}3x\text{}\text{if}-2<x\le 1\\ x+1\text{if}x>1\end{array}$.

Answer to Problem 37RE

Solution:

d. The range of the function is $(-\infty , \infty )$.

Explanation of Solution

Given:

$f\left(x\right)=\{\begin{array}{l}3x\text{}\text{if}-2<x\le 1\\ x+1\text{if}x>1\end{array}$

Calculation:

d. Range of the function:

From the graph, we find that the range is $f\left(x\right)>-6$ or $\left(-6,\infty \right)$.