https://file.aiquickdraw.com/mathbot/file/17283046362579121pfyr.jpg

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.

Let f(x) = sqrt(x^2 - 9) and g(x) = 2/x. Determine the domain of f + g, f/g, and g(f(x)).

Learn how to find the domain for the sum, division, and composition of functions f(x) = sqrt(x^2 - 9) and g(x) = 2/x. This problem covers key concepts like the domain of square root and rational functions and how to intersect domains for function operations. Suitable for Grades 10-12.

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