Let $f(x)=\frac{1}{x-2}$ and $g(x)=\frac{3}{x}+2$
Find the following functions. Simplify your answers.
\[
\begin{array}{l}
f(g(x))= \\
g(f(x))=
\end{array}
\]
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Step 1 ：Find the formula for \(f(g(x))\) and simplify your answer. The formula for \(f(g(x))\), also known as the composition of functions f and g, is given by \(f(g(x))\). This means that we substitute \(g(x)\) into the function \(f(x)\). In this case, \(g(x) = \frac{3}{x}+2\), so we substitute this into \(f(x) = \frac{1}{x-2}\) to get \(f(g(x)) = \frac{1}{(\frac{3}{x}+2)-2}\).

Step 2 ：Simplify the expression \(f(g(x)) = \frac{1}{(\frac{3}{x}+2)-2}\). First, simplify the denominator: \((\frac{3}{x}+2)-2 = \frac{3}{x}\). So, \(f(g(x)) = \frac{1}{\frac{3}{x}}\). Next, we can simplify this by multiplying the numerator and the denominator by x to get \(f(g(x)) = \frac{x}{3}\).

Step 3 ：\(\boxed{f(g(x)) = \frac{x}{3}}\)