Finding the Inverse Function Formula for Rational Expressions

[soroban]

One semester I was asked to find the inverse of $\,f(x) \:=\:\dfrac{3x - 5}{2x+1}$
Later, I had to find the inverse of $\,f(x) \:=\:\dfrac{2x+7}{4x-3}$

It occurred to me that a general formula would a handy tool.
Especially since I planned to teach Mathematics and I might
be teaching this very topic every semester.

So I solved it for: $\,f(x) \:=\:\dfrac{ax+b}{cx+d}$

And arrived at: $\:f^{\text{-}1}(x) \;=\;\dfrac{dx-b}{\text{-}cx+a}$

This is easily remembered . . .

(1) Switch the coefficients on the main diagonal ($a$ and $d$).

(2) Change the signs on the minor diagonal ($b$ and $c$).

 

[ineedhelpnow]

Nice! That's real cool, soroban :)