Questions: Find two nontrivial functions (f(x)) and (g(x)) so (f(g(x))=frac6(x+8)^4)

Solution Steps

Let $g(x) = x + 8$. This transformation shifts the input $x$ by 8, simplifying the expression we need to work with.

Let $f(x) = \frac{6}{x^4}$. This function is designed to take the output of $g(x)$ and produce the desired result when composed with $g(x)$.

Now, we compute the composition $f(g(x))$: $$f(g(x)) = f(x + 8) = \frac{6}{(x + 8)^4}$$ This matches the required expression $\frac{6}{(x + 8)^{4}}$, confirming that our choices for $f(x)$ and $g(x)$ are correct.

Final Answer