Question

consider the following relation.

( x = 5|y| - 5 )

step 2 of 2: find the domain and range of the inverse. express your answer in interval notation.

answer

domain =

range =

Explanation:

Step1: Find inverse of the relation

Swap $x$ and $y$:
$$y = 5|x| - 5$$
Solve for $x$:
$$y + 5 = 5|x|$$
$$|x| = \frac{y + 5}{5}$$
$$x = \pm\frac{y + 5}{5}$$
The inverse relation is $x = \pm\frac{y + 5}{5}$, or rewritten as $y = \pm\frac{x + 5}{5}$.

Step2: Find domain of inverse

For the inverse relation $y = \pm\frac{x + 5}{5}$, there are no restrictions on $x$ (all real numbers are allowed, since we can substitute any real $x$ into the expression).
Domain: $(-\infty, \infty)$

Step3: Find range of inverse

The original relation $x = 5|y| - 5$ has a range for $x$: since $|y| \geq 0$, $5|y| \geq 0$, so $x = 5|y| - 5 \geq -5$. The range of the original relation becomes the domain of the inverse, and the domain of the original relation (all real $y$) becomes the range of the inverse.
Range: $(-\infty, \infty)$

Answer:

Domain = $(-\infty, \infty)$
Range = $(-\infty, \infty)$