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The question asks about the relationship between two composed functions, \( f(g(x)) \) and \( g(f(x)) \), and which statement about their compositions is true.

### Possible answers:
1. \( f(g(x)) \) will **always** be the same as \( g(f(x)) \).
2. \( f(g(x)) \) will **never** be the same as \( g(f(x)) \).
3. \( f(g(x)) \) **could** be the same as \( g(f(x)) \), depending on the functions.

### Explanation:
- **Compositions of functions** \( f(g(x)) \) and \( g(f(x)) \) do not generally result in the same output. The order in which the functions are applied matters, so in most cases, \( f(g(x)) \neq g(f(x)) \).
  
- However, in **certain cases**, depending on the functions, \( f(g(x)) \) could equal \( g(f(x)) \). For example, if both \( f(x) \) and \( g(x) \) are identity functions (i.e., \( f(x) = x \) and \( g(x) = x \)), then their compositions will be equal.

### Correct Answer:
The correct choice is:
- \( f(g(x)) \) **could be the same** as \( g(f(x)) \), depending on the functions.

Would you like further details or any clarifications?

Here are 5 related questions to explore further:
1. What are some examples where \( f(g(x)) \) equals \( g(f(x)) \)?
2. Can the commutative property ever apply to function composition?
3. How does function composition relate to real-world applications?
4. What happens to the compositions if \( f(x) \) and \( g(x) \) are inverses of each other?
5. How can we prove whether \( f(g(x)) = g(f(x)) \) for specific functions?

**Tip:** Always verify the output of composed functions by carefully applying them in sequence!

Suppose you have two functions f(x) and g(x). Then which statement about their compositions is true?

Explore the relationship between composed functions f(g(x)) and g(f(x)) to determine when they might be the same. Learn about function composition, algebraic principles, and how the order of operations affects the result. Suitable for high school students.

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