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

1 Answer
May 14, 2017

Answer:

#(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#