What is the domain of the function f(x) = sqrt(9 - x^2)/(x + 2)?

To determine the domain of the function ( f(x) = \frac{\sqrt{9 - x^2}}{x + 2} ), we need to consider the constraints imposed by both the square root and the denominator.

First, let's examine the square root, ( \sqrt{9 - x^2} ). For the expression inside the square root to be non-negative, we require:

[ 9 - x^2 \geq 0. ]

This simplifies to:

[ x^2 \leq 9. ]

Taking the square root of both sides, we find:

[ -3 \leq x \leq 3. ]

Therefore, ( x ) must fall within the interval ([-3, 3]) to ensure that we have real output values from the square root function.

Next, we need to examine the denominator, ( x + 2 ). For the function to be defined, the denominator cannot be zero:

[ x + 2 \neq 0, ]

which implies:

[ x \neq -2. ]

Combining these two conditions, ( x ) must satisfy both the interval from the square root ( [-3, 3