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3-59.

What is $f^{−1}\left(x\right)$ if $f(x)=\frac{x}{x+1}$ ?

Step 1:

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

Step 2:

$\text{Switch }x\text{ and }y: x = \frac{y}{y + 1}$

Step 3:

Multiply both sides by (<i>y</i> + 1).

$(\textit{y} + 1)\textit{x} = \textit{y}$

Distribute.

$yx + x = y$

Isolate <i>y</i> on the left side.

$yx - y = -x$

Factor out <i>y</i>.

$y(x - 1) = -x$

Divide by sides by (<i>x</i> − 1).

$y = \frac{-\textit{x}}{\textit{x} - 1}\rightarrow\:\textit{y} = \frac{\textit{x}}{1-\textit{x}}$

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