Rather than use the f^(-1)(x) notation as inverse of f(x),
I find it easier to define another function g(x) as the inverse of f(x). This is easier to type and avoid the interpretation f^(-1)(x)=1/(f(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)