If f(x)=(2x)/(x-5), find f^-1(x) and state the domain of f^-1(x). I believe we start by substituting f(x) with y, and interchanging the variables, but I am lost after that step.

Expert Answers

An illustration of the letter 'A' in a speech bubbles

If `f(x)=(2x)/(x-5)` find `f^(-1)(x)` and the domain of `f^(-1)(x)` .

One method is to let `y=(2x)/(x-5)` , then exchange x and y, then solve for y. Thus:

`x=(2y)/(y-5)` exchange x and y
`x(y-5)=2y`
`xy - 5x=2y`
`xy-2y=5x`
`y(x-2)=5x`
`y=(5x)/(x-2)` This is a new function which is the inverse
of the given function.

We exchange x and y because the input of the inverse function is the output of the original function and vice versa. In other words, the domain of the inverse function is the range of the original function.

Thus `f(x)=(2x)/(x-5) => f^(-1)(x)=(5x)/(x-2)` . The domain of `f^(-1)(x)` is `x!= 2` .

** Note that the range of the original function is `y != 2` as there is a horizontal asymptote at y=2. The domain of the original function is `x != 5` , and the range of the inverse function is `y != 5` **

Approved by eNotes Editorial Team

We’ll help your grades soar

Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.

  • 30,000+ book summaries
  • 20% study tools discount
  • Ad-free content
  • PDF downloads
  • 300,000+ answers
  • 5-star customer support
Start your 48-Hour Free Trial