The function you have provided is:

$f(x) = \sqrt{\frac{x-1}{x^2 - 9}}$

Domain of the Function

The function involves a square root and a division, so its domain is determined by:

1. The square root: The expression inside the square root, $\frac{x-1}{x^2 - 9}$, must be non-negative: $\frac{x-1}{x^2 - 9} \geq 0$
2. The denominator: The term $x^2 - 9$ must not be zero: $x^2 - 9 \neq 0 \quad \implies \quad x \neq \pm 3$

Step-by-Step Analysis

1. Solve $\frac{x-1}{x^2 - 9} \geq 0$:

The inequality is satisfied where: $x-1 \geq 0 \quad \text{and} \quad x^2 - 9 > 0 \quad \text{or} \quad x^2 - 9 < 0$

(a) Critical points: Solve for zeros:

• $x-1 = 0 \implies x = 1$
• $x^2 - 9 = 0 \implies x = \pm 3$

(b) Intervals to test: Divide the real number line into intervals based on these critical points: $(-\infty, -3)$, $(-3, 1)$, $(1, 3)$, and $(3, \infty)$.

2. Test the sign of $\frac{x-1}{x^2-9}$ in each interval:

$\text{Signs of } (x-1) \text{ and } (x^2-9):$

• For $x \in (-\infty, -3)$: $x-1 < 0$, $x^2 - 9 > 0 \implies \frac{x-1}{x^2-9} < 0$
• For $x \in (-3, 1)$: $x-1 < 0$, $x^2 - 9 < 0 \implies \frac{x-1}{x^2-9} > 0$
• For $x \in (1, 3)$: $x-1 > 0$, $x^2 - 9 < 0 \implies \frac{x-1}{x^2-9} < 0$
• For $x \in (3, \infty)$: $x-1 > 0$, $x^2 - 9 > 0 \implies \frac{x-1}{x^2-9} > 0$

3. Combine with restrictions:

• From the square root, $\frac{x-1}{x^2 - 9} \geq 0$, valid intervals are $(-3, 1)$ and $(3, \infty)$.
• Exclude $x = \pm 3$ since $x^2 - 9 = 0$ there.

Thus, the domain of $f(x)$ is: $x \in (-3, 1] \cup (3, \infty)$

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Would you like a detailed graph or further analysis?

Relative Questions:

1. How can we graph this function to better understand its behavior?
2. What happens to the function as $x \to \pm \infty$?
3. Can we calculate the derivative $f'(x)$ for further analysis of the function?
4. What are the asymptotes (if any) of this function?
5. How do we solve inequalities involving square roots in general?

Tip:

When solving inequalities with fractions, test intervals around critical points to determine the signs of the numerator and denominator.