How do you find the inverse of f(x)=(x-1)/(x+2)f(x)=x1x+2?

1 Answer
Nov 24, 2015

If f^(-1)(x)f1(x) is the inverse of f(x)f(x)
then f^-(1)(x)=(2x+1)/(1-x)f(1)(x)=2x+11x

Explanation:

Rather than use the f^(-1)(x)f1(x) notation as inverse of f(x)f(x),
I find it easier to define another function g(x)g(x) as the inverse of f(x)f(x). This is easier to type and avoid the interpretation f^(-1)(x)=1/(f(x))f1(x)=1f(x). Fell free to replace all my g(x)'s with f^(-1)(x)#.

Let g(x) be the inverse of f(x)

By definition of inverse:
color(white)("XXX")f(g(x))=x

But given f(x)=(x-1)/(x+2) it follows that
color(white)("XXX")f(g(x)) = (g(x)-1)/(g(x)+2)

Therefore
color(white)("XXX")(g(x)-1)/(g(x)+2) = x

color(white)("XXX")g(x)-1 = g(x)*x + 2x

color(white)("XXX")(g(x)-g(x)*x)=2x+1

color(white)("XXX")g(x)(1-x) = 2x+1

color(white)("XXX")g(x)= (2x+1)/(1-x)