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

To find the inverse function \( f^{-1}(x) \) of \( f(x) = \frac{x}{x+1} \), we start by letting \( y = f(x) \): \[ y = \frac{x}{x+1} \] Next, we solve for \( x \) in terms of \( y \). We begin by cross-multiplying: \[ y(x + 1) = x \] Expanding this gives: \[ yx + y = x \] Next, we move all terms involving \( x \) to one side: \[ yx - x = -y \] Factoring out \( x \) from the left side: \[ x(y - 1) = -y \] Now, we solve for \( x \): \[ x = \frac{-y}{y - 1} \] This represents the inverse function. To express it more clearly, we switch \( y \) back to \( x \): \[ f^{-1}(x) = \frac{-x}{x - 1} \] Thus, the inverse function is: \[ \boxed{f^{-1}(x) = \frac{-x}{x - 1}} \]