Chapter 4 - Exponential and Logarithmic Functions - Section 4.7 Financial Models - 4.7 Assess Your Understanding - Page 349: 78

$f^{-1}(x)=\dfrac{2 x}{x-1}$

Let $f(x)=y \Rightarrow x=f^{-1}(y)$ $\Rightarrow \quad y=\frac{x}{x-2}$ $\Rightarrow \quad y(x-2)=x$ $\Rightarrow \quad x y-2 y=x$ $\Rightarrow x y-x=2 y$ $\Rightarrow \quad x(y-1)=2 y$ $\Rightarrow \quad x=\frac{2 y}{y-1}$ Replace $y$ with $x$ and $x$ with $y$ to obtain: $y=\dfrac{2x}{x-1}$ Replace $y$ with $f^{-1}(x)$ to obtain: $f^{-1}(x)=\dfrac{2 x}{x-1}$