Questions: Find the domain of the function. f(x) = √(8+x) / (8-x) Write your answer as an interval or union of intervals.

Solution Steps

The function $f(x) = \frac{\sqrt{8+x}}{8-x}$ has two restrictions:

1. The expression under the square root must be non-negative: $8 + x \geq 0$.
2. The denominator cannot be zero: $8 - x \neq 0$.

To ensure the square root is defined: $$8 + x \geq 0 \implies x \geq -8.$$

To ensure the denominator is not zero: $$8 - x \neq 0 \implies x \neq 8.$$

The domain of $f(x)$ is all real numbers $x$ such that $x \geq -8$ and $x \neq 8$. This can be written as the interval $[-8, 8) \cup (8, \infty)$.

Final Answer