Me i can't figure this one

f(x)= 2/x-6

g(x)= 6x+2/x

determine whether or not f(x) and g(x) are inverses of each other.

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

i can't see the picture : /