Question

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

Answer 1

Domain: `{x | x in RR}`
Range: `{y | y le 2}`

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 `RR`.

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 RR` 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]}