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

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

Transcript text: Find two nontrivial functions $f(x)$ and $g(x)$ so $f(g(x))=\frac{6}{(x+8)^{4}}$

Solution

Solution Steps

Step 1: Define $ g(x) $

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

Step 2: Define $ f(x) $

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) $.

Step 3: Compute the Composition

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

\[ \begin{array}{l} f(x) = \frac{6}{x^4} \\ g(x) = x + 8 \end{array} \]

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