Question

Find the domain of the function $$f(x) = \frac{1}{x+8} + \frac{1}{\sqrt{x - 8}} + \frac{1}{\sqrt{8 - x}}.$$ Express your answer as a union of intervals.

Answer 1

For the function to be defined, the denominators of all the fractions must be nonzero. The first fraction has a nonzero denominator for all $x \neq -8,$ the second fraction has a nonzero denominator for all $x \neq 8,$ and the third fraction has a nonzero denominator for all $x \neq 8.$ Therefore, the function is defined for all $x$ such that
\[x \neq -8 \quad \text{and} \quad x \neq 8.\]The solution is thus $\boxed{(-\infty,-8) \cup (-8,8) \cup (8,\infty)}.$