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Let's analyze the given problem step-by-step.

The functions are:
- \( f(x) = \frac{1}{x^2} \)
- \( g(x) = \frac{1}{9x} \)

### (a) Domain of \( f(x) \)

The function \( f(x) = \frac{1}{x^2} \) has a denominator of \( x^2 \), and it cannot be zero because division by zero is undefined. Therefore, we need to exclude \( x = 0 \) from the domain.

Thus, the domain of \( f(x) \) is:
\[
\boxed{(-\infty, 0) \cup (0, \infty)}
\]

### (b) Domain of \( g(x) \)

The function \( g(x) = \frac{1}{9x} \) has a denominator of \( 9x \). Again, we cannot divide by zero, so \( x = 0 \) must be excluded from the domain.

Thus, the domain of \( g(x) \) is:
\[
\boxed{(-\infty, 0) \cup (0, \infty)}
\]

### (c) Domain of \( f \circ g \) (Composition \( f(g(x)) \))

The composition \( f(g(x)) \) means we are applying \( f \) to \( g(x) \). Therefore, we need to find the domain where \( g(x) \) is defined and where \( f(g(x)) \) is also defined.

First, we know that \( g(x) \) is undefined at \( x = 0 \), so \( x = 0 \) must be excluded.

Now, we need to check the domain of \( f(g(x)) \). Since \( f(x) = \frac{1}{x^2} \), we need to ensure that \( g(x) \neq 0 \) when substituted into \( f \). 
We solve:
\[
g(x) = \frac{1}{9x} = 0
\]
This equation has no solutions, meaning \( g(x) \) is never zero for any \( x \neq 0 \).

Therefore, the domain of \( f \circ g \) is the same as the domain of \( g(x) \), excluding \( x = 0 \):
\[
\boxed{(-\infty, 0) \cup (0, \infty)}
\]

Would you like further details on these steps, or do you have any questions? Here are 5 related questions for more practice:

1. What is the range of \( f(x) = \frac{1}{x^2} \)?
2. Can you find the range of \( g(x) = \frac{1}{9x} \)?
3. How would you graph \( f(x) \) and \( g(x) \) to verify their domains?
4. What happens to the domain of \( f \circ g \) if the function definitions change slightly?
5. Can you determine the range of the composite function \( f \circ g \)?

**Tip:** When finding the domain of a composite function, always start with the inner function and ensure its values are valid for the outer function.

Determine the domains of f(x) = 1/x^2 and g(x) = 1/(9x), and find the domain of f(g(x)) using interval notation.

Learn how to determine the domains of the functions f(x) = 1/x^2, g(x) = 1/(9x), and their composition f(g(x)) using interval notation. This detailed solution covers key concepts in function domains and composite functions.

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