Chapter 2.4, Problem 32AYU

In Problems 31-42:

(a) Find the domain of each function.

(b) Locate any intercepts.

(c) Graph each function.

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

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

To determine

To find: The following values of the function $f\left(x\right)=\{\begin{array}{c}x+3 if x<-2\\ -2x-3 if x\ge -2\end{array}$

a. Domain of the function $f$.

b. Intercepts of the function $f$ if any.

c. Graph of the function $f$.

d. Range of the function $f$ based on its graph.

Answer to Problem 32AYU

Solution:

a. Domain of the function $f$: $\text{The set of all real numbers}\text{. That is }\left(-\infty ,\infty \right)$.

b. Intercepts of the function $f$ if any: $x\text{-intercepts are }\left(-3,0\right)\text{ and }\left(-1.5,0\right)\text{ and }y\text{-intercept is }(0,-3)$.

c. Graph of the function $f$:

d. Range of the function $f$ based on its graph: The range takes the real values which are lesser than or equal to $-2$. $\text{Range }=\left\{x/x\le -2,x\in R\right\}$.

Explanation of Solution

Given:

The function is: $f\left(x\right)=\{\begin{array}{c}x+3 if x<-2\\ -2x-3 if x\ge -2\end{array}$

It is asked to find the domain and intercepts of the function $f$. Also, sketch the function and conclude its range from it.

a. Domain of the function $f$: To find the domain of $f$, look at its definition. Since $f$ is defined for all $x<\mathrm{-2}$ and $x\ge \mathrm{-2}$, the domain of $f$ is the set of all real numbers. That is, $\left(-\infty ,\infty \right)$.

b. Intercepts: The $y\text{-intercept}$ of the graph of the function is $f(0)$. Because the equation for $f$ when $x=0$ is $f\left(x\right)=-2x-3$, the $y\text{-intercept}$ is $f\left(0\right)=-2\left(0\right)-3=\mathrm{-3}$.

The $x\text{-intercepts}$ of the graph of a function $f$ are the real solutions to the equation $f\left(x\right)=\text{ 0}$. To find the $x\text{-intercepts}$ of $f$, solve $f\left(x\right)=\text{ 0}$ for each “piece” of the function, and then determine which values of $x$, if any, satisfy the condition that defines the piece.

$f\left(x\right)=\text{ 0}$

$x+3=0$

$x=-3$

Satisfies the condition $x<\mathrm{-2}$.

$f\left(x\right)=\text{ 0}$

$-2x-3=0$

$x=\frac{3}{2}$

Satisfies the condition $x\ge -2$.

Therefore, the $x\text{-intercepts are }\left(-3,0\right)\text{ and }\left(-1.5,0\right)$.

c.

d. From the graph, it can be easily predicted that the range takes the real values which are lesser than or equal to $-2$. $\text{Range = }\left\{\frac{x}{x}\le -2,x\in R\right\}$