Let's go step by step to evaluate $f(g(x))$ when $f(x) = \frac{1}{x+2}$ and $g(x) = \frac{x}{x-8}$.

Step 1: Substitute $g(x)$ into $f(x)$

To find $f(g(x))$, we substitute the expression for $g(x)$ into $f(x)$. This means wherever there is an $x$ in the function $f(x)$, we replace it with $g(x)$.

$f(g(x)) = \frac{1}{g(x) + 2}$

Step 2: Replace $g(x)$ with $\frac{x}{x-8}$

Now, substitute the expression for $g(x)$ into the equation for $f(g(x))$:

$f(g(x)) = \frac{1}{\frac{x}{x-8} + 2}$

Step 3: Simplify the expression

We need to simplify the denominator. First, express the "2" as a fraction with a common denominator:

$f(g(x)) = \frac{1}{\frac{x}{x-8} + \frac{2(x-8)}{x-8}}$ $f(g(x)) = \frac{1}{\frac{x + 2(x-8)}{x-8}}$ Now simplify the numerator:

$x + 2(x-8) = x + 2x - 16 = 3x - 16$

So the expression becomes:

$f(g(x)) = \frac{1}{\frac{3x - 16}{x-8}}$

Step 4: Simplify further

Now, simplify the overall expression:

$f(g(x)) = \frac{1}{\frac{3x - 16}{x-8}} = \frac{x - 8}{3x - 16}$

Final Answer:

Thus, the simplified expression for $f(g(x))$ is:

$f(g(x)) = \frac{x - 8}{3x - 16}$

This explains why the expression $\frac{x - 8}{3x - 16}$ is listed as the correct answer.

Would you like any further clarification or more details?

Related Questions:

1. How do you evaluate composite functions like $f(g(x))$ for more complex functions?
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