Diketahui f(x)=3x+2 dan g(x)=-2x+k . Jika (fog)(x)=(gof)(x)

(x - 8) / 9

Pembahasan

Diketahui:
f(x) = 3x + 2
g(x) = -2x + k
(fog)(x) = (gof)(x)

(fog)(x) = f(g(x)) = f(-2x + k) = 3(-2x + k) + 2 = -6x + 3k + 2
(gof)(x) = g(f(x)) = g(3x + 2) = -2(3x + 2) + k = -6x - 4 + k

Karena (fog)(x) = (gof)(x):
-6x + 3k + 2 = -6x - 4 + k
3k + 2 = -4 + k
3k - k = -4 - 2
2k = -6
k = -3

Maka, g(x) = -2x - 3.

Selanjutnya kita cari (fof)(x):
(fof)(x) = f(f(x)) = f(3x + 2) = 3(3x + 2) + 2 = 9x + 6 + 2 = 9x + 8

Untuk mencari inversnya, misalkan y = 9x + 8
y - 8 = 9x
x = (y - 8) / 9
Jadi, (fof)^(-1)(x) = (x - 8) / 9.