The domain of f(x) is the set of all real values except 7, and the domain of g(x) is the set of all real values except –3. Which of the following describes the domain

When determining the domain of a composite function such as \( f(g(x)) \) or \( g(f(x)) \), the domain will be restricted by both functions. Specifically:

1. **For \( f(g(x)) \):** The domain consists of all \( x \) values for which \( g(x) \) is within the domain of \( f(x) \). This means:
   - \( g(x) \) must be in the domain of \( f(x) \), which means \( g(x) \neq 7 \).
   - \( x \) must be in the domain of \( g(x) \), which means \( x \neq -3 \).

2. **For \( g(f(x)) \):** The domain consists of all \( x \) values for which \( f(x) \) is within the domain of \( g(x) \). This means:
   - \( f(x) \) must be in the domain of \( g(x) \), which means \( f(x) \neq -3 \).
   - \( x \) must be in the domain of \( f(x) \), which means \( x \neq 7 \).

**Therefore, the domain of \( f(g(x)) \) is the set of all real numbers except where \( x = -3 \) or where \( g(x) = 7 \).**

**Similarly, the domain of \( g(f(x)) \) is the set of all real numbers except where \( x = 7 \) or where \( f(x) = -3 \).**

### Summary:
- The domain of \( f(g(x)) \) is the set of all real numbers except \( x = -3 \) and any \( x \) where \( g(x) = 7 \).
- The domain of \( g(f(x)) \) is the set of all real numbers except \( x = 7 \) and any \( x \) where \( f(x) = -3 \).

Would you like to dive into details or have any questions? Here are 5 related questions you might find useful:

1. What happens if both functions \( f(x) \) and \( g(x) \) have the same excluded values in their domains?
2. How can we find the explicit expressions for \( f(g(x)) \) and \( g(f(x)) \)?
3. Can you explain how to determine the range of a composite function?
4. What happens to the domain if we only know one of the functions?
5. How does the domain change if we add, subtract, or multiply two functions?

**Tip:** Always check the domain of the inner function first when dealing with composite functions.

The domain of f(x) is the set of all real values except 7, and the domain of g(x) is the set of all real values except –3. Which of the following describes the domain?

Explore how to determine the domain of composite functions f(g(x)) and g(f(x)) when given the domains of f(x) and g(x). This problem covers advanced concepts in functions and is suitable for students in Grades 11-12.

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