How do you find the domain and range of y=-abs(x)+2?

Oct 7, 2017

Domain: $\left\{x | x \in \mathbb{R}\right\}$
Range: $\left\{y | y \le 2\right\}$

Explanation:

The domain is all numbers that $x$ could be without making $y$ undefined.

In this case, there isn't any value that will make $y$ undefined, since there are no fractions with $x$ in the denominator or functions with undefined values ($| x |$ is defined for all real numbers). Therefore, the domain of this function is all real numbers, or $\mathbb{R}$.

The range is every value that $y$ could be for the values of $x$ in the domain.

We know that the range of $y = | x |$ is $y \ge 0$, since the absolute value function returns only positive numbers, or 0 if the input is 0.

This means that the range of $y = - | x |$ is $y \le 0$, since we're taking every value in the range and making it negative.

This means that the range of $y = - | x | + 2$ is $y \le 2$, since we're adding $2$ to every value in the range.

Therefore, the domain is $x \in \mathbb{R}$ and the range is $y \le 2$.

This is what the graph of this function looks like (notice that all values of $x$ have a point at some $y$ value, and all values of $y$ less than or equal to $2$ have a point at some $x$ value):
graph{y = -(abs(x))+2 [-10, 10, -5, 5]}