Problem 9

Explanation

To find the domain of the function, we need to ensure that all components of the function are defined (values inside square roots must be non-negative, and the denominator cannot be zero).

The function $f(x) = \frac{\sqrt{5 - x}}{\sqrt{x^2 + x - 6}}$ has two main conditions:

Condition 1: Numerator

The value inside the square root in the numerator must be non-negative:

Condition 2: Denominator

The value inside the square root in the denominator must be positive (since the square root in the denominator cannot be zero):

The zeros of this inequality are $x = -3$ and $x = 2$.

Using test points, we find the solution region for the denominator: $x < -3$ or $x > 2$.

Domain Intersection

The domain of function $f$ is the intersection of the two conditions above:

1. $x \leq 5$
2. $x < -3$ or $x > 2$

The intersection is:

• For $x < -3$, it satisfies $x \leq 5$.
• For $x > 2$, it is bounded by $x \leq 5$, becoming $2 < x \leq 5$.

Therefore, the domain of function $f$ is: