How To Find The Domain Of An Absolute Value Function

Absolute Value Functions

An absolute value function is a function that contains an algebraic expression within absolute value symbols. Recall that the absolute value of a number is its distance from $0$ on the number line.

The absolute value parent function, written as $f (x) = | x |$ , is defined as

$f (x) = {\begin{cases} x if x > 0 \\ 0 if x = 0 \\ - x if x < 0 \end{cases}$

To graph an absolute value function, choose several values of $x$ and find some ordered pairs.

$x$	$y = \| x \|$
−2	2
−1	1
0	0
1	1
2	2

Plot the points on a coordinate plane and connect them.

Observe that the graph is V-shaped.

( $1$ ) The vertex of the graph is $(0, 0)$ .

( $2$ ) The axis of symmetry ( $x = 0$ or $y$ -axis) is the line that divides the graph into two congruent halves.

( $3$ ) The domain is the set of all real numbers.

( $4$ ) The range is the set of all real numbers greater than or equal to $0$ . That is, $y \geq 0$ .

( $5$ ) The $x$ -intercept and the $y$ -intercept are both $0$ .

Vertical Shift

To translate the absolute value function $f (x) = | x |$ vertically, you can use the function

$g (x) = f (x) + k$ .

When $k > 0$ , the graph of $g (x)$ translated $k$ units up.

When $k < 0$ , the graph of $g (x)$ translated $k$ units down.

Horizontal Shift

To translate the absolute value function $f (x) = | x |$ horizontally, you can use the function

$g (x) = f (x - h)$ .

When $h > 0$ , the graph of $f (x)$ is translated $h$ units to the right to get $g (x)$ .

When $h < 0$ , the graph of $f (x)$ is translated $h$ units to the left to get $g (x)$ .

Stretch and Compression

The stretching or compressing of the absolute value function $y = | x |$ is defined by the function $y = a | x |$ where $a$ is a constant. The graph opens up if $a > 0$ and opens down when $a < 0$ .

For absolute value equations multiplied by a constant $(for example, y = a | x |)$ ,if $0 < a < 1$ , then the graph is compressed, and if $a > 1$ , it is stretched. Also, if a is negative, then the graph opens downward, instead of upwards as usual.

More generally, the form of the equation for an absolute value function is $y = a | x - h | + k$ . Also:

The vertex of the graph is $(h, k)$ .
The domain of the graph is set of all real numbers and the range is $y \geq k$ when $a > 0$ .
The domain of the graph is set of all real numbers and the range is $y \leq k$ when $a < 0$ .
The axis of symmetry is $x = h$ .
It opens up if $a > 0$ and opens down if $a < 0$ .
The graph $y = | x |$ can be translated $h$ units horizontally and $k$ units vertically to get the graph of $y = a | x - h | + k$ .
The graph $y = a | x |$ is wider than the graph of $y = | x |$ if $| a | < 1$ and narrower if $| a | > 1$ .