Chapter 3.7, Problem 52E

To determine

ToGraph:The function $y=\left|x\right|-3$ . Then rewrite the absolute value function as two linear functions, one that has the domain $x<0$ and one that has the domain $x\ge 0$

Answer to Problem 52E

The graph of the given function is

  

For the domain $x<0$ , the absolute value function $y=\left|x\right|-3$ can be written as the linear function as $y=-x-3$

And, for the domain $x\ge 0$ , the absolute value function $y=\left|x\right|-3$ can be written as the linear function as $y=x-3$

Explanation of Solution

Given:

The function $y=\left|x\right|-3$

Calculation:

For thefunction $y=\left|x\right|-3$

Calculating the values of the function at various $x\text{-values}$

At $x=-5$

  $\begin{array}{c}y=\left|-5\right|-3\\ =5-3\\ =2\end{array}$

At $x=-3$

  $\begin{array}{c}y=\left|-3\right|-3\\ =3-3\\ =0\end{array}$

At $x=0$

  $\begin{array}{c}y=\left|0\right|-3\\ =0-3\\ =-3\end{array}$

At $x=2$

  $\begin{array}{c}y=\left|2\right|-3\\ =2-3\\ =-1\end{array}$

At $x=5$

  $\begin{array}{c}y=\left|5\right|-3\\ =5-3\\ =2\end{array}$

Making the table of values of $x$ and $y$

$x$	$-5$	$-3$	$0$	$2$	$5$
$y$	$2$	$0$	$-3$	$-1$	$2$

Plotting the points on the graph and joining the points, the graph of the given function is

  

For the given absolute value function $y=\left|x\right|-3$

Since,

  $\begin{array}{c}y=\left|x\right|-3\\ =\{\begin{array}{l}-x-3x<0\\ x-3x\ge 0\end{array}\end{array}$

Therefore, for the domain $x<0$ , the absolute value function $y=\left|x\right|-3$ can be written as the linear function as $y=-x-3$

And, for the domain $x\ge 0$ , the absolute value function $y=\left|x\right|-3$ can be written as the linear function as $y=x-3$

Conclusion:

The graph of the given function is

  

For the domain $x<0$ , the absolute value function $y=\left|x\right|-3$ can be written as the linear function as $y=-x-3$

And, for the domain $x\ge 0$ , the absolute value function $y=\left|x\right|-3$ can be written as the linear function as $y=x-3$