Chapter 4, Problem 54SGR

To determine

To Graph :

Graph the function and state the domain and range.

Answer to Problem 54SGR

Domain is all real numbers.

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

Explanation of Solution

Given Information:

Given equation is :

  $y=f(x)=|2x-2|$

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

  $\begin{array}{l}f(x)=|2x-2|=0\\ \Rightarrow f(x)=\pm (2x-2)=0\\ \Rightarrow 2x-2=0\\ \Rightarrow 2x-2+2=0+\mathrm{2.......................}[adding"2"each\_side]\\ \Rightarrow 2x=2\\ \Rightarrow \frac{2x}{2}=\frac{2}{2}\mathrm{......................................}[divide"2"each\_side]\\ \Rightarrow x=1\end{array}$

Make a table for different value in the domain :

  $y=f(x)=|2x-2|=\{\begin{array}{l}2x-2;2x-2>0\\ -(2x-2);2x-2\le 0\end{array}=\{\begin{array}{l}2x-2;x>1\\ -(2x-2);x\le 1\end{array}$

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

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