Find $$ f^{-1} $$ if $$ f(x)=\frac{x}{x+1} $$

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To find the inverse function $$ f^{-1} $$ of the given function $$ f(x) = \frac{x}{x+1} $$, we need to follow these steps:

1. Replace $$ f(x) $$ with $$ y $$ in the given function.
2. Swap $$ x $$ and $$ y $$ in the equation.
3. Solve the resulting equation for $$ y $$ to find the inverse function.

Let's start by replacing $$ f(x) $$ with $$ y $$ in the given function:

$$ y = \frac{x}{x+1} $$

Now, we swap $$ x $$ and $$ y $$ in the equation:

$$ x = \frac{y}{y+1} $$

Next, we solve the resulting equation for $$ y $$ to find the inverse function:

$$ x(y+1) = y $$

$$ xy + x = y $$

$$ xy - y = -x $$

$$ y(x-1) = -x $$

$$ y = \frac{-x}{x-1} $$

Therefore, the inverse function $$ f^{-1} $$ is given by:

$$ f^{-1}(x) = \frac{-x}{x-1} $$
The inverse function is $$ f^{-1}(x) = \frac{-x}{x-1} $$.

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