Chapter 4.7, Problem 20PPS

To determine

To Graph :

Graph the function and state the domain and range.

Answer to Problem 20PPS

Domain is all real numbers.

Range is non-negative real numbers = $\{y:y\ge 0\}$

Explanation of Solution

Given Information:

Given equation is :

  $y=g(x)=|-x-3|$

Concept Used :

Absolute Value Function:

An absolute value function is a function that contains an algebraic expression within absolute value symbols. The absolute value of a number is its distance from 0 on the number line. Parent Function : $f(x)=\left|x\right|=\{\begin{array}{l}x;x>0\\ -x;x\le 0\end{array}$

Domain : All real numbers

Range : All non-negative real numbers

Graph :

  

Calculation :

Since g(x) cannot be negative , so the minimum point of the graph will be where g(x)= 0.

  $\begin{array}{l}g(x)=|-x-3|=0\\ \Rightarrow g(x)=\pm (-x-3)=0\\ \Rightarrow -x-3=0\\ \Rightarrow -x-3+3=0+\mathrm{3......................}[adding"3"each\_side]\\ \Rightarrow -x=3\\ \Rightarrow \frac{-x}{-1}=\frac{3}{-1}\mathrm{.................................}[divide"-1"each\_side]\\ \Rightarrow x=-3\end{array}$

Make a table for different value in the domain :

  $y=g(x)=|-x-3|=\{\begin{array}{l}-x-3;-x-3>0\\ -(-x-3);-x-3\le 0\end{array}=\{\begin{array}{l}-x-3;x<-3\\ -(-x-3);x\ge -3\end{array}$

x	-x-3	$y=g(x)=|-x-3|$
-5	2	2
-4	1	1
-3	0	0
-2	-1	1
-1	-2	2

Plotting the values and joining, we get the required graph :

  

The x -coordinate values specify the domain of the function . Since the graph covers all possible values of x , the domain is all real numbers.

The y -coordiante values specify the range of the function . Since , the graph does not take any value less than y =0 or any negative value , the range is = $\left\{y:y\ge 0\right\}$