Question

Graph each function. State the domain and range.

$\mathit{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathbf{|}\mathbf{2}\mathbf{x}\mathbf{-}\mathbf{2}\mathbf{|}$

Short answer

The domain is all real numbers and the range is non-negative real numbers i.e.,$\mathbf{\{}\mathbf{y}\mathbf{:}\mathbf{y}\mathbf{\ge }\mathbf{0}\mathbf{\}}$.

Step-by-step solution

Step 1. State the concept for 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.

$f\left(x\right)=\left|x\right|=\left\{\begin{array}{c}x;x>0\\ -x;x\le 0\end{array}\right\}$

Step 2. Solve for x.

Since $f\left(x\right)$cannot be negative, so the minimum point of the graph will be where $f\left(x\right)=0$.

$\begin{array}{c}f\left(x\right)=|2x-2|\\ f\left(x\right)=\pm (2x-2)=0\\ 2x-2=0\\ 2x=2\\ x=1\end{array}$

Step 3. Make a table for the given function.

Construct a table for f(x) by taking some values of $x$.

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

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

Step 4. Plot the graph.

Plot the values of $x$and $f\left(x\right)$on a 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 $\{y:y\ge 0\}$.