# What is the domain and range for y= -abs(x-5)?

Jun 25, 2018

See below.

#### Explanation:

There is no restriction on $x$, so domain is:

$\left\{x \in \mathbb{R}\right\}$

or

$\left(- \infty , \infty\right)$

By definition of absolute value:

$| x - 5 | \ge 0$

Therefore:

$- | x - 5 | \le 0$

From this we can see that the minimum value is:

as $x \to \pm \infty$, $\textcolor{w h i t e}{8888} - | x - 5 | \to - \infty$

For $x = 5$

$| x - 5 | = 0$

This is the maximum value:

Range is therefore:

$\left\{y \in \mathbb{R} | - \infty < y \le 0\right\}$

or

$\left(- \infty , 0\right]$

The graph of $y = - | x - 5 |$ confirms this:

graph{y=-|x-5| [-1, 10, -5, 5]}