Question

How do you simplify `(x-3)/(x+2)`?

Answer 1

`(x-3)/(x+2) = 1-5/(x+2)`

Explanation:

As a rational expression,

`(x-3)/(x+2)`

is already in simplest form.

However, it is possible to separate it out into the sum of a trivial polynomial and a partial fraction with constant numerator:

`(x-3)/(x+2) = ((x+2)-5)/(x+2) = 1-5/(x+2)`

One advantage of this form is that `x` only occurs once, so it is easy to invert as a function:

Let:

`y = 1-5/(x+2)`

Add `5/(x+2)-y` to both sides to get:

`5/(x+2) = 1-y`

Take the reciprocal of both sides to get:

`(x+2)/5 = 1/(1-y)`

Multiply both sides by `5` to get:

`x+2 = 5/(1-y)`

Subtract `2` from both sides to get:

`x = 5/(1-y)-2`

So if:

`f(x) = (x-3)/(x+2) = 1-5/(x+2)`

then:

`f^(-1)(y) = 5/(1-y)-2`