The problem in the image states the following:

1. Let $f(x) = \sqrt{x^2 - 9}$ and $g(x) = \frac{2}{x}$. Determine the domain.
   • a. $f + g$
   • b. $\frac{f}{g}$
   • c. $g \circ f$, then determine the formula of $g \circ f$.

Let's analyze these functions and solve each part.

Step 1: Finding the domain of $f(x) = \sqrt{x^2 - 9}$

The expression inside the square root must be non-negative: $x^2 - 9 \geq 0$ $(x - 3)(x + 3) \geq 0$ This inequality holds for $x \leq -3$ or $x \geq 3$. So, the domain of $f(x)$ is: $(-\infty, -3] \cup [3, \infty)$

Step 2: Finding the domain of $g(x) = \frac{2}{x}$

The denominator must not be zero: $x \neq 0$ Thus, the domain of $g(x)$ is: $(-\infty, 0) \cup (0, \infty)$

Part (a): Domain of $f + g$

The domain of $f + g$ is the intersection of the domains of $f$ and $g$: $\text{Domain of } f + g = ((-\infty, -3] \cup [3, \infty)) \cap ((-\infty, 0) \cup (0, \infty))$ This results in: $(-\infty, -3] \cup [3, \infty)$

Part (b): Domain of $\frac{f}{g}$

For the fraction $\frac{f}{g}$ to be defined, both $f(x)$ and $g(x)$ must be defined, and $g(x) \neq 0$. Since $g(x) = \frac{2}{x}$, it is already clear that $x \neq 0$. The domain remains: $(-\infty, -3] \cup [3, \infty)$

Part (c): Domain and formula of $g \circ f$

To find the domain of the composition $g \circ f$, we need $f(x)$ to be in the domain of $g$. Since $g(x) = \frac{2}{x}$, we need $f(x) \neq 0$. The values for $f(x)$ must be positive or negative but not zero. Since $f(x) = \sqrt{x^2 - 9}$, it only equals zero when $x = \pm 3$, which are the boundaries of its domain.

Thus, the domain of $g \circ f$ remains: $(-\infty, -3] \cup [3, \infty)$

The formula for $g \circ f$ is: $g(f(x)) = \frac{2}{\sqrt{x^2 - 9}}$

Would you like more details or have any questions about these solutions?

Here are five related questions to deepen your understanding:

1. What would be the domain if the function were $f(x) = \sqrt{9 - x^2}$?
2. How do you solve quadratic inequalities in general?
3. What happens to the domain of a composition $f \circ g$ if $g(x)$ has a smaller domain than $f(x)$?
4. What are the general conditions for adding or dividing two functions in terms of their domains?
5. How would the range of these functions affect their compositions?

Tip: When finding the domain of a composition of functions, always check that the output of the inner function lies within the domain of the outer function.