Question

How do you find the domain of `f(x)= sqrt(x+6)/(x-5)`?

Answer 1

For this function to have a value we need to satisfy two conditions:
(a) an expression under the sign of a square root should not be negative, that is `x+6>=0`;
(b) an expression in the denominator should not be equal to `0`, that is `x-5!=0`.

The first condition resolves into
(a) `x >= -6`
The second one resolves into
(b) `x != 5`

Therefore, the domain of `f(x)=sqrt(x+6)/(x-5)` consists of two segments:
`-6 <= x <5` and `x > 5`