Question

Determine the domain and range of the function graphed below.

Short answer

The domain is all real numbers and the range is $\left\{y:y\ge -2\right\}$.

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. State the concept of Domain and Range.

The domain is the set of all possible $x$-values that will make the function "work" and will output real $y$-values. Determine the DOMAIN of each function by looking for those values of the independent variable (usually $x$) which are allowed to use. (Usually, avoid 0 on the bottom of a fraction, or negative values under the square root sign).

The range is the resulting $y$-values after substituting all the possible $x$-values. The RANGE of a function is the spread of possible $y$-values (minimum $y$-value to maximum $y$-value). Substitute different $x$-values into the expression for $y$to see what is happening. Make sure for minimum and maximum values of $y$.

Step 3. Make a table for the given function.

Construct a table for $f\left(x\right)=\left|x\right|$by taking some values of $x$.

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

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 $y$-coordinate values specify the range of the function. Since, the graph does not take any value less than $y=-2$, the range is $\left\{y:y\ge -2\right\}$.