Home > PC3 > Chapter 3 > Lesson 3.1.3 > Problem3-47

3-47.

Write the equation of $f^{-1}(x)$ if:

1. $f\left(x\right)=\frac{x}{x+1}$

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

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

Multiply both sides by $(y+1)$.
$(\textit{y} + 1)\textit{x} = \textit{y}$

$yx + x = y$

Isolate $y$ on the left side.
$yx - y = -x$

Factor out $y$.
$y(x - 1) = -x$

Divide both sides by$\left(x-1\right)$.
$y = \frac{-\textit{x}}{\textit{x} - 1}\rightarrow\:\textit{y} = \frac{\textit{x}}{1-\textit{x}}$

1. $f\left(x\right)=5\sqrt[3]{x-2}$