Question

What is the domain and range of `y=-|x|-9`?

Answer 1

Domain: `x inℝ` (all real numbers)

Range: `y<=-9`

Explanation:

The domain of the function `y=-|x|-9` is all real numbers because any number plugged in for `x` yields a valid output `y`.

Since there is a minus sign in front of the absolute value, we know that the graph "opens downward," like this:
graph{|x|*-1 [-10, 10, -5, 5]}
(This is the graph of `-|x|`.)

This means that the function has a maximum value. If we find the maximum value, we can say that the function's range is `y<=n`, where `n` is that maximum value.

The maximum value can be found by graphing the function:

graph{|x|*-1-9 [-10, 10, -15, -5]}

The highest value that the function reaches is `-9`, so this is the maximum value. Finally, we can say that the range of the function is `y<=-9`.