two functions f(x) and g(x) are called inverse of each other when these follow this rule.
if we put x in f we get value f(x)
then putting f(x) in g should give us value x
in other word
let's say I put x = a in f(x) we get f(a)
now when we will put x = f(a) in g(x) we should get a back.
then they called inverse of each other.
let's take an example
f(x) = 4x-2
g(x) = (x+2)/4
let's put x = 1 in f(x)
f(1) = 4×1-2 = 4-2 = 2
f(1)= 2
now we will put 2 in g(x) and it should give us value 1
g(f(1))= g(2) =(2+2)/4= 4/4 = 1
yes we got 1. doing same in reverse order . that is we will check now g(x) first.
lets take x= -2 for simplification
g(-2) = (-2+2)/4 = 0
now f(0) = 4×0-2 = -2
yes we got it again so in this case functions are inverse to each other.
mathematically
g(f(x)) = x = f(g(x))
let's prove that if above function follows this or not .
f(x) = 2/x - 6
g(x) = 6x + 2/x
let's calculate g(f(x)) first
g(f(x)) = g(2/x -6) = 6(2/x -6) + 2/(2/x -6)
= 12/x -36 + 2x/(2-6x)
which is not equal to x so we don't have to proceed further. they are not inverse of each other