Chapter 4.7, Problem 6CYU

To determine

To Graph :

Graph the function and state the domain and range.

Answer to Problem 6CYU

Domain is all real numbers.

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

Explanation of Solution

Given Information:

Given equation is :

  $y=g(x)=|2x+4|$

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.

  $\begin{array}{l}g(x)=|2x+4|=0\\ \Rightarrow g(x)=\pm (2x+4)=0\\ \Rightarrow 2x+4=0\\ \Rightarrow 2x+4-4=0-\mathrm{4.......................}[subtracting"4"each\_side]\\ \Rightarrow 2x=-\mathrm{4......................................}[simplify]\\ \Rightarrow \frac{2x}{2}=\frac{-4}{2}\mathrm{....................................}[divide"2"each\_side]\\ \Rightarrow x=-2\end{array}$

Make a table for different value in the domain :

  $y=g(x)=|2x+4|=\{\begin{array}{l}2x+4;2x+4>0\\ -(2x+4);2x+4\le 0\end{array}=\{\begin{array}{l}2x+4;x>-2\\ -(2x+4);x\le -2\end{array}$

x	2x+4	$y=g(x)=|2x+4|$
-3	-2	2
-2	0	0
-1	2	2
0	4	4
1	6	6

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