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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Evaluate the expression by following steps:
- step0: Evaluate:
  $$\frac{x}{x-3}$$
- step1: Substitute:
  $$\frac{\frac{2}{x}}{\frac{2}{x}-3}$$
Substitute $$ x=\frac{x}{x-3} $$ into the expression $$ \frac{2}{x} $$.
Evaluate the expression by following steps:
- step0: Evaluate:
  $$\frac{2}{x}$$
- step1: Substitute:
  $$\frac{2}{\frac{x}{x-3}}$$
Calculate or simplify the expression $$ \frac{2}{\frac{x}{x-3}} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{2}{\frac{x}{x-3}}$$
- step1: Multiply by the reciprocal:
  $$2\times \frac{x-3}{x}$$
- step2: Multiply the terms:
  $$\frac{2\left(x-3\right)}{x}$$
- step3: Simplify:
  $$\frac{2x-6}{x}$$
Calculate or simplify the expression $$ \frac{\frac{2}{x}}{\frac{2}{x}-3} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\frac{2}{x}}{\frac{2}{x}-3}$$
- step1: Subtract the terms:
  $$\frac{\frac{2}{x}}{\frac{2-3x}{x}}$$
- step2: Multiply by the reciprocal:
  $$\frac{2}{x}\times \frac{x}{2-3x}$$
- step3: Reduce the fraction:
  $$2\times \frac{1}{2-3x}$$
- step4: Multiply the terms:
  $$\frac{2}{2-3x}$$
Let's find the compositions of the functions $$ f(x) $$ and $$ g(x) $$.

1. **Finding $$ f(g(x)) $$**:
   - We start with $$ g(x) = \frac{2}{x} $$.
   - Substitute $$ g(x) $$ into $$ f(x) $$:
     $$
     f(g(x)) = f\left(\frac{2}{x}\right) = \frac{\frac{2}{x}}{\frac{2}{x} - 3}
     $$
   - Now, simplify this expression:
     $$
     f(g(x)) = \frac{\frac{2}{x}}{\frac{2}{x} - 3} = \frac{2}{2 - 3x}
     $$

2. **Finding $$ g(f(x)) $$**:
   - We start with $$ f(x) = \frac{x}{x-3} $$.
   - Substitute $$ f(x) $$ into $$ g(x) $$:
     $$
     g(f(x)) = g\left(\frac{x}{x-3}\right) = \frac{2}{\frac{x}{x-3}}
     $$
   - Now, simplify this expression:
     $$
     g(f(x)) = \frac{2}{\frac{x}{x-3}} = \frac{2(x-3)}{x} = \frac{2x - 6}{x}
     $$

Thus, the simplified results are:
- $$ f(g(x)) = \frac{2}{2 - 3x} $$
- $$ g(f(x)) = \frac{2x - 6}{x} $$
$$ f(g(x)) = \frac{2}{2 - 3x} $$ $$ g(f(x)) = \frac{2x - 6}{x} $$

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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