MHB Finding the Inverse Function Formula for Rational Expressions

AI Thread Summary
The discussion focuses on finding the inverse function formula for rational expressions, specifically for the form f(x) = (ax + b) / (cx + d). The author successfully derived the inverse as f^(-1)(x) = (dx - b) / (-cx + a). This formula simplifies the process of finding inverses, making it a useful tool for teaching mathematics. The method involves switching the coefficients on the main diagonal and changing the signs on the minor diagonal. Such a general formula aids in efficiently solving inverse problems in rational expressions.
soroban
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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 $c$).
 
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Nice! That's real cool, soroban :)
 
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