Question

How to find domain for `f(x) = sqrt(25 - x^2)`?

Answer 1

`[-5,5]`

Explanation:

So that the answer won't be imaginary, `25-x^2` must be greater than or equal to `0`.

`{x|25-x^2>=0}`
`{x|25-x^2-25>=0-25}`
`{x|-x^2>=-25}`
`{x|(-1)(-x^2)<=(-1)(-25)}`
`{x|x^2<=25}`

This will then be separated to two cases:
If `x` is positive:
`{x|x^2<=25}`
`{x|x<=5}`
`color(blue)((-oo,5])`

If `x` is negative:
`{x|x^2<=25}`
`{x|x>=-5}`
`color(red)([-5,oo))`

We will then get the intersection of the two intervals:
`color(blue)((-oo,5])nncolor(red)([-5,oo))`
`color(magenta)([-5,5]`