# Inverse function question

• Apr 10th 2010, 12:22 PM
Iceflash234
Inverse function question
Suppose $f(x)$ coincides with its inverse over a suitable interval.

Suppose that $f(x) = \frac{ax+b}{cx+d}$. What conditions on a,b,c,d are necessary and sufficient in order that $f(x)$ coincide with it's inverse function?

Help is greatly appreciated. Thanks.
• Apr 10th 2010, 12:25 PM
$f^{-1}(x) = {b-dx\over cx-a}$

btw, I think you meant $ax+b\over cx + \color{red}d$.
• Apr 10th 2010, 12:38 PM
Iceflash234
Quote:

$f^{-1}(x) = {b-dx\over cy-a}$

btw, I think you meant $ax+b\over cx + \color{red}d$.

Thanks. Fixed the typo.

About response: Ok. I understand that. But what conditions on a,b,c,d are are necessary and sufficient?
• Apr 10th 2010, 12:46 PM
Equate $f(x) = f^{-1}(x)$ and clear the denominators. You will arrive at two polynomials in x. Equating their coefficients should give you the required conditions.