Chapter 4.7, Problem 19PPS

To determine

To Graph :

Graph the function and state the domain and range.

Answer to Problem 19PPS

Domain is all real numbers.

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

Explanation of Solution

Given Information:

Given equation is :

  $y=g(x)=|-3x-5|$

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.

  $\begin{array}{l}g(x)=|-3x-5|=0\\ \Rightarrow g(x)=\pm (-3x-5)=0\\ \Rightarrow -3x-5=0\\ \Rightarrow -3x-5+5=0+\mathrm{5......................}[adding"5"each\_side]\\ \Rightarrow -3x=5\\ \Rightarrow \frac{-3x}{-3}=\frac{5}{-3}\mathrm{.................................}[divide"-3"each\_side]\\ \Rightarrow x=-\frac{5}{3}\end{array}$

Make a table for different value in the domain :

  $y=g(x)=|-3x-5|=\{\begin{array}{l}-3x-5;-3x-5>0\\ -(-3x-5);-3x-5\le 0\end{array}=\{\begin{array}{l}-3x-5;x>-\frac{5}{3}\\ -(-3x-5);x\le -\frac{5}{3}\end{array}$

x	-3x-5	$y=g(x)=|-3x-5|$
-3	4	4
-2	1	1
$-\frac{5}{3}$	0	0
-1	-2	2
0	-5	5

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 = $\left\{y:y\ge 0\right\}$