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