Chapter 2.4, Problem 34AYU

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.

$f\left(x\right)=\{\begin{array}{l}2x+5\text{if}-3\le x<0\\ -3\text{if}x=0\\ -5x\text{if}x>0\end{array}$

To determine

To find: The following values of the function, $f\left(x\right)=\{\begin{array}{l}2x+5 if -3\le x<0\\ -3 if x=0\\ -5x if x>0\end{array}$

a. Domain of the function $f$.

Answer to Problem 34AYU

Solution:

a. Domain of the function $f$: $\left[-3,\infty \right)$.

Explanation of Solution

Given:

The function is: $f\left(x\right)=\{\begin{array}{l}2x+5 if -3\le x<0\\ -3 if x=0\\ -5x if x>0\end{array}$

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 $-3\le x<0,x=0\text{and} x>0$, the domain of $f$ is $\left[-3,\infty \right)$.

To determine

To find: The following values of the function, $f\left(x\right)=\{\begin{array}{l}2x+5 if -3\le x<0\\ -3 if x=0\\ -5x if x>0\end{array}$

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

Answer to Problem 34AYU

Solution:

b. Intercepts of the function $f$ if any: $x-\text{intercept is }(-2.5,0)\text{ and }y\text{-intercept is }(0,-3)$.

Explanation of Solution

Given:

The function is: $f\left(x\right)=\{\begin{array}{l}2x+5 if -3\le x<0\\ -3 if x=0\\ -5x if x>0\end{array}$

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

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

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

$f\left(x\right)=0$

$2x+5=0$

$x=-5/2$

Satisfies the condition $-3\le x<0$.

$f\left(x\right)=0$

$-3=0$

Contradicts the condition $x=0$.

$f\left(x\right)=0$

$-5x=0$

$x=0$

Contradicts the condition $x>0$.

Therefore, the $x\text{-intercept is }(-5/2,0)$.

To determine

To find: The following values of the function, $f\left(x\right)=\{\begin{array}{l}2x+5 if -3\le x<0\\ -3 if x=0\\ -5x if x>0\end{array}$

c. Graph of the function $f$.

Answer to Problem 34AYU

Solution:

c. Graph of the function $f$:

Explanation of Solution

Given:

The function is: $f\left(x\right)=\{\begin{array}{l}2x+5 if -3\le x<0\\ -3 if x=0\\ -5x if x>0\end{array}$

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

c.

To determine

To find: The following values of the function, $f\left(x\right)=\{\begin{array}{l}2x+5 if -3\le x<0\\ -3 if x=0\\ -5x if x>0\end{array}$

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

Answer to Problem 34AYU

Solution:

d. Range of the function $f$ based on its graph: The range takes the real number 5 and values which are lesser than or equal to 4. $\text{Range }=\left(5,-\infty \right)$.

Explanation of Solution

Given:

The function is: $f\left(x\right)=\{\begin{array}{l}2x+5 if -3\le x<0\\ -3 if x=0\\ -5x if x>0\end{array}$

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

d. Range of the function $f$ based on its graph: The range takes the real number 5 and values which are lesser than or equal to 4. $\text{Range }=\left(5,-\infty \right)$.