Chapter 11.6, Problem 21E

To determine

To explain:

The inverse of the following function is function or not.

  $f\left(x\right)=\frac{1}{2x-1}$

Answer to Problem 21E

This graph passes the horizontal line test, thus the function has inverse that is the function.

Explanation of Solution

Given information:

  $f\left(x\right)=\frac{1}{2x-1}$

The given function is:

  $f\left(x\right)=\frac{1}{2x-1}$

Replace the $f\left(x\right)$ by $y$ :

  $y=\frac{1}{2x-1}$

Switch $x$ and $y$ :

  $x=\frac{1}{2y-1}$

Determine $y$ as a function of $x$ :

  $\begin{array}{l}x\left(2y-1\right)=1\\ 2xy-x=1\\ 2xy=x+1\\ y=\frac{x+1}{2x}\end{array}$

Replace $y$ by $f^{-1}\left(x\right)$ :

  $f^{-1}\left(x\right)=\frac{x+1}{2x}$

Graph the function and then perform the horizontal line test:

  

Graph: $f\left(x\right)=\frac{1}{2x-1}$

Therefore, this graph passes the horizontal line test, thus the function has inverse that is the function.