Tentukan f^(-1)(x), g^(-1)(x),(fog)^(-1)(x), dan

f^-1(x) = x/(1-x), g^-1(x) = (x-3)/2, (fog)^-1(x) = (4x-3)/(2-2x), (gof)^-1(x) = (x-3)/(5-x).

Pembahasan

Diketahui: f(x) = x/(x+1) dan g(x) = 2x+3

1. Mencari f^-1(x):
Misalkan y = f(x) = x/(x+1)
y(x+1) = x
yx + y = x
y = x - yx
y = x(1 - y)
x = y / (1 - y)
Jadi, f^-1(x) = x / (1 - x)

2. Mencari g^-1(x):
Misalkan y = g(x) = 2x + 3
y - 3 = 2x
x = (y - 3) / 2
Jadi, g^-1(x) = (x - 3) / 2

3. Mencari (fog)^-1(x) = (g^-1 o f^-1)(x):
(fog)^-1(x) = g^-1(f^-1(x))
= g^-1(x / (1 - x))
= [(x / (1 - x)) - 3] / 2
= [(x - 3(1 - x)) / (1 - x)] / 2
= (x - 3 + 3x) / (2(1 - x))
= (4x - 3) / (2 - 2x)

4. Mencari (gof)^-1(x) = (f^-1 o g^-1)(x):
(gof)^-1(x) = f^-1(g^-1(x))
= f^-1((x - 3) / 2)
= [(x - 3) / 2] / [1 - (x - 3) / 2]
= [(x - 3) / 2] / [(2 - (x - 3)) / 2]
= (x - 3) / (2 - x + 3)
= (x - 3) / (5 - x)