Question

Demuestre que ( f ᵒ g )(x) = ( g ᵒ f )(x) = x a) sabiendo que f(x) = 2x y g(x) = x/2 b) sabiendo que f(x) = 2x – 6 y g(x) = (x + 6)/2

Answer 1

The answer is the demonstration, which is in the step-by-step explanation.

Explanation:

Composite functions:

( f ᵒ g )(x) = f(g(x))

( g ᵒ f )(x) = g(f(x))

a)

f(x) = 2x

g(x) = x/2

`f(g(x)) = f((x)/(2)) = 2(x)/(2) = x`

`g(f(x)) = g(2x) = (2x)/(2) = x`

Then

`f(g(x)) = g(f(x)) = x`

b)

f(x) = 2x - 6

g(x) = (x + 6)/2

`f(g(x)) = f((x+6)/(2)) = 2(x+6)/(2) - 6 = x + 6 - 6 = x`

`g(f(x)) = g(2x - 6) = (2x - 6 + 6)/(2) = (2x)/(2) = x`

Then

`f(g(x)) = g(f(x)) = x`