Question

How do you find the domain of `h(x)= sqrt( 4-x^2)/( x-3)`?

Answer 1

The domain is restricted by the numerator to `[-2, 2]` and by the denominator to `(-oo,3) uu (3,oo)`, which is no additional restriction. So the domain of `h(x)` is `[-2, 2]`

Explanation:

In order for the numerator of `h(x)` to be defined, `4 - x^2 >= 0`.

Adding `x^2` to both sides we get `x^2 <= 4 = 2^2`

hence `-2 <= x <= 2`

In order for the denominator to be defined and non-zero, we just need `x != 3`

This condition only affects values of `x` outside the `[-2, 2]` range we already know the domain is restricted to.

So `h(x)` is well defined for all `x in [-2, 2]`.