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!