Find $$ f^{-1}(x) $$ for the function. $$ f(x)=\frac{x}{x+4} $$ $$ f^{-1}(x)= $$

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To find the inverse function $$ f^{-1}(x) $$ for the function $$ f(x) = \frac{x}{x+4} $$, we will follow these steps:

1. Replace $$ f(x) $$ with $$ y $$:
   $$
   y = \frac{x}{x+4}
   $$

2. Swap $$ x $$ and $$ y $$:
   $$
   x = \frac{y}{y+4}
   $$

3. Solve for $$ y $$:
   - Multiply both sides by $$ y + 4 $$:
   $$
   x(y + 4) = y
   $$
   - Distribute $$ x $$:
   $$
   xy + 4x = y
   $$
   - Rearrange the equation to isolate $$ y $$:
   $$
   xy - y = -4x
   $$
   - Factor out $$ y $$:
   $$
   y(x - 1) = -4x
   $$
   - Solve for $$ y $$:
   $$
   y = \frac{-4x}{x - 1}
   $$

Thus, the inverse function is:
$$
f^{-1}(x) = \frac{-4x}{x - 1}
$$

Now, let's confirm this result by outputting the final expression.

Final result:
$$
f^{-1}(x) = \frac{-4x}{x - 1}
$$
$$ f^{-1}(x) = \frac{-4x}{x - 1} $$

Find for the function.

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