Final answer:
The expression f(f^{-1}(f(x))) simplifies to f(x) because applying the inverse function f^{-1} to f(x) results in the original input x, and then applying f to x gives f(x) again.
Therefore, f(f^{-1}(f(x))) equals 3x + 6.
Step-by-step explanation:
To solve for f(f^{-1}(f(x))), we need to understand the properties of the function f and its inverse f^{-1}.
The function f is given as f(x) = 3x + 6.
The inverse function f^{-1}(x) is the function that, when applied to f(x), returns the original input x.
Since applying f^{-1} to f(x) should give us x, when we apply f to that result, we should once again obtain f(x).
Specifically, f(f^{-1}(f(x))) simplifies to f(x).
Therefore, no matter what input x we choose, as long as f(x) is defined and f and its inverse f^{-1} are proper inverses, the result will always be f(x), which in this case is 3x + 6.