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