Question

What is the domain and range of `y = - sqrt(9-x^2)`?

Answer 1

Domain: `[-3, 3]`
Range: `[-3, 0]`

Explanation:

In order to find the function's domain, you need to take into account the fact that, for real numbers, you can only take the square root of a positive number.

In other words, in oerder for the function to be defined, you need the expression that's under the square root to be positive.

`9 - x^2 >= 0`

`x^2 <= 9 implies |x| <= 3`

This means that you have

`x >= -3" "` and `" "x<=3`

For any value of `x` outside the interval `[-3, 3]`, the expression under the square root will be negative, which means that the function will be undefined. Therefore, the domain of the function will be `x in [-3, 3]`.

Now for the range. For any value of `x in [-3, 3]`, the function will be negative.

The maximum value the expression under the radical can take is for `x=0`

`9 - 0^2 = 9`

which means that the minimum value of the function will be

`y = -sqrt(9)= -3`

Therefore, the range of the function will be `[-3, 0]`.

graph{-sqrt(9-x^2) [-10, 10, -5, 5]}