Question

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

Answer 1

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

Explanation:

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)`