Questions: Given f(x), find g(x) and h(x) such that f(x)=g(h(x)) and neither g(x) nor h(x) is solely x. f(x)=3/(3x+4)

Transcript text: Given $f(x)$, find $g(x)$ and $h(x)$ such that $f(x)=g(h(x))$ and neither $g(x)$ nor $h(x)$ is solely $x$. \[ f(x)=\frac{3}{3 x+4} \] Answer

Solution

Solution Steps

Step 1: Identify $h(x)$

Given the function $f(x) = \frac{3}{3x + 4}$, we identify $h(x) = bx + c$, which is the linear part of the denominator of $f(x)$. Substituting the parameters, we get $h(x) = 3x + 4$.

Step 2: Set $g(x)$

Next, we set $g(x) = \frac{3}{x}$. This choice of $g(x)$ ensures that when composed with $h(x)$, we retrieve the original function $f(x)$.

Step 3: Verify that $g(h(x)) = f(x)$

To verify, we compose $g(x)$ with $h(x)$: $g(h(x)) = \frac{3}{(3x + 4)} = 3/(3x + 4)$, which matches the original function $f(x)$.

Final Answer:

The decomposition of $f(x) = \frac{3}{3x + 4}$ is such that $h(x) = 3x + 4$ and $g(x) = 3/x$. This demonstrates that $f(x) = g(h(x))$, completing the decomposition.

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