Let $$ f(x)=\frac{x}{x-3} $$ and $$ g(x)=\frac{2}{x} $$ Find the following functions. Simplify your answers. $$ f(g(x))=\square $$ $$ g(f(x))=\square $$

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1. To find $$ f(g(x)) $$, substitute $$ g(x) $$ into $$ f(x) $$:
   $$
   f(g(x)) = f\left(\frac{2}{x}\right) = \frac{\frac{2}{x}}{\frac{2}{x} - 3}
   $$

2. Rewrite the denominator using a common denominator:
   $$
   \frac{2}{x} - 3 = \frac{2 - 3x}{x}
   $$

3. Now, substitute back to get:
   $$
   f(g(x)) = \frac{\frac{2}{x}}{\frac{2-3x}{x}} = \frac{2}{2-3x}
   $$

4. Next, to find $$ g(f(x)) $$, substitute $$ f(x) $$ into $$ g(x) $$:
   $$
   g(f(x)) = g\left(\frac{x}{x-3}\right) = \frac{2}{\frac{x}{x-3}}
   $$

5. Simplify by multiplying by the reciprocal:
   $$
   g(f(x)) = 2 \cdot \frac{x-3}{x} = \frac{2(x-3)}{x}
   $$

Thus, the simplified functions are:
$$
f(g(x)) = \frac{2}{2-3x} \quad \text{and} \quad g(f(x)) = \frac{2(x-3)}{x}
$$
$$ f(g(x)) = \frac{2}{2-3x} $$ and $$ g(f(x)) = \frac{2(x-3)}{x} $$.

Let \( f(x)=\frac{x}{x-3} \) and \( g(x)=\frac{2}{x} \) Find the following functions. Simplify your answers. \( f(g(x))=\square \) \( g(f(x))=\square \)

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