Chapter 2.4, Problem 33AYU

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.

$f\left(x\right)=\{\begin{array}{l}x+3\text{if}-2\le x<1\\ 5\text{if}x=1\\ -x+2\text{if}x>1\end{array}$

To determine

To find:The following values of the function, $f\left(x\right)=\{\begin{array}{l}x+3 if -2\le x<1\\ 5 if x=1\\ -x+2 if x>1\end{array}$

a. Domain of the function $f$ .

Answer to Problem 33AYU

Solution:

a. Domain of the function $f:\left[-2,\infty \right)$.

Explanation of Solution

Given:

The function is: $f\left(x\right)=\{\begin{array}{l}x+3 if -2\le x<1\\ 5 if x=1\\ -x+2 if x>1\end{array}$

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 $-2\le x<1,x=1\text{and} x>1$, the domain of $f$ is $\left[-2,\infty \right)$.

To determine

To find:The following values of the function, $f\left(x\right)=\{\begin{array}{l}x+3 if -2\le x<1\\ 5 if x=1\\ -x+2 if x>1\end{array}$

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

Answer to Problem 33AYU

Solution:

b. Intercepts of the function $f$ if any: $x\text{-intercept is }(2,0)\text{ and }y\text{-intercept is }(0,3)$.

Explanation of Solution

Given:

The function is: $f\left(x\right)=\{\begin{array}{l}x+3 if -2\le x<1\\ 5 if x=1\\ -x+2 if x>1\end{array}$

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 $f\left(x\right)=x+3$, the $y\text{-intercept}$ is $f\left(x\right)=0+3=3$.

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.

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

$x+3=0$

$x=-3$

Contradicts the condition $-2\le x<1$.

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

$5=0$

Contradicts the condition $x=1$.

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

$-x+2=0$

$x=2$

Satisfies the condition $x>1$

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

To determine

To find:The following values of the function, $f\left(x\right)=\{\begin{array}{l}x+3 if -2\le x<1\\ 5 if x=1\\ -x+2 if x>1\end{array}$

c. Graph of the function $f$.

Answer to Problem 33AYU

Solution:

c. Graph of the function $f$:

Explanation of Solution

Given:

The function is: $f\left(x\right)=\{\begin{array}{l}x+3 if -2\le x<1\\ 5 if x=1\\ -x+2 if x>1\end{array}$

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, $f\left(x\right)=\{\begin{array}{l}x+3 if -2\le x<1\\ 5 if x=1\\ -x+2 if x>1\end{array}$

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

Answer to Problem 33AYU

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. $\text{Range }=\left\{5\right\}\cup \left[4,-\infty \right)$.

Explanation of Solution

Given:

The function is: $f\left(x\right)=\{\begin{array}{l}x+3 if -2\le x<1\\ 5 if x=1\\ -x+2 if x>1\end{array}$

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. $\text{Range }=\left\{5\right\}\cup \left[4,-\infty \right)$.