Under which conditions does the commutative property apply to compositions of f(x) and g(x)?

when f(g(x)) and g(f(x) are inverse functions
when f(g(x)) and g(f(x) are equal for at least one value of x
when f(g(x)) and g(f(x) are equal for all values of x
when f(g(x)) and g(f(x) are not inverse functions

The commutative property applies to compositions of functions \( f(x) \) and \( g(x) \) **when \( f(g(x)) \) and \( g(f(x)) \) are equal for all values of \( x \)**. This means that for any input \( x \), applying \( g(x) \) first and then \( f(x) \), and vice versa, yields the same result.

Thus, the correct condition is:

- **When \( f(g(x)) \) and \( g(f(x)) \) are equal for all values of \( x \)**.

### Would you like more details or further explanation?

Here are 5 related questions that expand on this concept:
1. What is the difference between commutative and associative properties in function composition?
2. Can the commutative property apply to some values of \( x \) but not all?
3. How can we verify if two functions are commutative for composition?
4. Does the commutative property hold for addition and multiplication of functions?
5. How does the commutative property of function composition relate to inverse functions?

**Tip:** To check if \( f(g(x)) = g(f(x)) \), compute both compositions explicitly and compare their results for all \( x \).

Under which conditions does the commutative property apply to compositions of f(x) and g(x)?

Explore when the commutative property applies to compositions of f(x) and g(x). This question focuses on function composition and how f(g(x)) and g(f(x)) relate. Learn about inverse functions and key conditions for equality in function compositions. Ideal for students in Grades 10-12.

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