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Let's break down the problem based on the image provided:

The given functions are:
- \( f(x) = \frac{4}{|x|} \)
- \( g(x) = x - 3 \)

You are asked to determine the domains of the following:
(a) The domain of \( f(x) \).
(b) The domain of \( g(x) \).
(c) The domain of \( f \circ g(x) \), which is the composition \( f(g(x)) \).

### (a) Domain of \( f(x) = \frac{4}{|x|} \):
- The function \( f(x) \) involves an absolute value in the denominator. For \( \frac{4}{|x|} \) to be defined, the denominator cannot be zero.
- Thus, \( x \neq 0 \).
- The domain of \( f(x) \) is all real numbers except zero: \( (-\infty, 0) \cup (0, \infty) \).

### (b) Domain of \( g(x) = x - 3 \):
- \( g(x) \) is a linear function with no restrictions on \( x \). It is defined for all real numbers.
- Therefore, the domain of \( g(x) \) is \( (-\infty, \infty) \).

### (c) Domain of \( f \circ g(x) = f(g(x)) \):
- To find the domain of the composition \( f(g(x)) \), first substitute \( g(x) = x - 3 \) into \( f(x) \):
  \[
  f(g(x)) = \frac{4}{|x - 3|}
  \]
- Similar to \( f(x) \), the denominator \( |x - 3| \) cannot be zero. This means \( x - 3 \neq 0 \), or \( x \neq 3 \).
- Thus, the domain of \( f(g(x)) \) is all real numbers except \( x = 3 \): \( (-\infty, 3) \cup (3, \infty) \).

### Final Answers:
- (a) Domain of \( f(x) \): \( (-\infty, 0) \cup (0, \infty) \)
- (b) Domain of \( g(x) \): \( (-\infty, \infty) \)
- (c) Domain of \( f \circ g(x) \): \( (-\infty, 3) \cup (3, \infty) \)

Would you like more details on any of these points?

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Here are 5 related questions to expand on this:

1. What would happen to the domain of \( f(x) \) if we changed it to \( f(x) = \frac{4}{x^2} \)?
2. How would the domain change if \( g(x) = \frac{1}{x-3} \)?
3. How does the composition of two functions generally affect the domain?
4. Could there be a case where \( f(g(x)) \) is undefined for an interval of values instead of just specific points?
5. What is the domain of \( f(g(x)) \) if \( f(x) = \frac{1}{x+2} \) and \( g(x) = \frac{1}{x} \)?

**Tip**: When dealing with rational functions, always check for points where the denominator equals zero to determine domain restrictions.

Determine the domains of the following functions based on the given functions: f(x) = 4/|x| and g(x) = x - 3.

Explore how to determine the domains of the functions f(x) = 4/|x| and g(x) = x - 3, and find the domain of the composition f(g(x)) = 4/|x - 3|. This involves identifying points where the functions are undefined due to zero denominators. Suitable for Grades 11-12 and college-level math.

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