Question

Diketahui f(x) = 10x-6 dan g(x) = 3-2x. Tentukan
a. (G o G^-1) (x) dan (G^-1 o G) (x)
b. (F o F^-1 o G) (x) dan (G o F^-1 o G^-1) (x)

Answer 1

F(x) = 10x - 6 --> f⁻¹(x) = 1/10 (x+ 6)
g(x) = 3 - 2x  --> g⁻¹(x) = -1/2 (x-3)

a) (gog⁻¹)(x) = g(g⁻¹) = 3 - 2 {-1/2 (x-3)}
gog⁻¹(x)= 3 + x - 3 = x

b) g⁻¹og (x)= g⁻¹(g) = -1/2 ( 3-2x -3)
g⁻¹og (x) = -1/2 (-2x) = x

c) (fof⁻¹ og) (x) =
 (fof⁻¹)(x) = f( f⁻¹) = 10 ( 1/10 (x+6))- 6 = x + 6 - 6 = x
fof⁻¹ (g(x)) =  {fof⁻¹ (g(x))} = g(x)= 3-2x

d) (gof⁻¹og⁻¹)(x)= ...
gof⁻¹ = g (f⁻¹) = 3 - 2 (1/10 (x+6)) = 3 - 1/5 (x+6)
gof⁻¹og⁻¹ (gof⁻¹)(g(x)) = 3 - 1/5 (3- 2x +6)
gof⁻¹og⁻¹(x)= 3 - 3/5 + 2/5 x - 6/5 
gof⁻¹og⁻¹(x) = 1/5 (15-3 + 2x -6) = 1/5 (2x + 6)