Question

Fungsi f:R→R dan g:R→R ditentukan oleh f(x)=2x+3 dan g(x)=x+1/(x-2) dengan x≠2. Tentukan :
a. (g o f)^-1(x) dan (f o g)^-1(x)
b. (g o f)^-1(2) dan (f o g)^-1(2)

Answer 1

F(x) = 2x +3 --> f⁻¹(x) = (x - 3)/2
g(x) = (x+1)/(x-2) --> g⁻¹(x) = (2x+ 1)/(x-1)

a) (gof)⁻¹(x) = f⁻¹ o g⁻¹ (x) = { 2x+1)/(x-1) - 3}/2
= ((2x-1)-3(x-1)} / 2(x-1)
= (2x-1- 3x + 3) /(2x - 2)
= (-x +2) /(2x -2)

(fog)⁻¹ (x)= g⁻¹ o f⁻¹(x)
g⁻¹{(x-3)/2} = { 2(x-3)/2 + 1 } / {(x-3)/2) -1}
= (4(x-3) + 2)/((x-3) - 2}
= (4x - 12 + 2) / (x -3 - 2)
= (4x - 10)/(x- 5)

b.
(gof)⁻¹(2) = (-2+2)/(4-2)= 0
(fog)⁻¹(2)= (4(2) -10 )/(2-5)=  - 2/-3 = 2/3