Question

How do you find the domain and range of `sqrt(25-x^2)`?

Answer 1

The domain is `= [-5,5]`
The range is `=[0,5]`

Explanation:

What is under the `sqrt` sign is `>=0`

Therefore,

`25-x^2>=0`

`(5+x)(5-x)>=0`

Let `f(x)=(5+x)(5-x)`

We can now build a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-5` `color(white)(aaaa)` `5` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `5+x` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `5-x` `color(white)(aaaaa)` `+` `color(white)(aaaa)` `+` `color(white)(aaaa)` `-`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `-`

Therefore,

`f(x)>=0` when `x in [-5, 5]`

The domain of `f(x)` is `D_f(x)= [-5,5]`

When `x=+-5`, `sqrt(f(x))=0`

When `x=0`, `sqrt(f(x))=5`, this is the max. value

so,
The range is `R = [0,5]`