Fungsi f: R -> R dan g: R -> R dirumuskan dengan

a. (f o g)^(-1)(x) = (2-3x)/(x-1), (g o f)^(-1)(x) = 1/(4-x), (f o f)^(-1)(x) = (x-1)/x, (g o g)^(-1)(x) = x-6. b. (f o g)(5) = 7/8, (g o f)^(-1)(5) = -1, (f o f)^(-1)(5) = 4/5, (g o g)^(-1)(5) = -1.

Pembahasan

a. Menentukan invers dari komposisi fungsi:

   1.  **(f o g)(x):**
       (f o g)(x) = f(g(x)) = f(x+3) = ((x+3)-1)/(x+3) = (x+2)/(x+3)
       Untuk mencari inversnya, misalkan y = (x+2)/(x+3).
       y(x+3) = x+2
       xy + 3y = x+2
       xy - x = 2 - 3y
       x(y-1) = 2 - 3y
       x = (2 - 3y) / (y-1)
       Jadi, (f o g)^(-1)(x) = (2 - 3x) / (x-1)

   2.  **(g o f)(x):**
       (g o f)(x) = g(f(x)) = g((x-1)/x) = ((x-1)/x) + 3 = (x-1 + 3x)/x = (4x-1)/x
       Untuk mencari inversnya, misalkan y = (4x-1)/x.
       xy = 4x - 1
       xy - 4x = -1
       x(y-4) = -1
       x = -1 / (y-4) = 1 / (4-y)
       Jadi, (g o f)^(-1)(x) = 1 / (4-x)

   3.  **(f o f)(x):**
       (f o f)(x) = f(f(x)) = f((x-1)/x) = (((x-1)/x) - 1) / ((x-1)/x)
       = ((x-1 - x)/x) / ((x-1)/x) = (-1/x) / ((x-1)/x) = -1 / (x-1)
       Untuk mencari inversnya, misalkan y = -1 / (x-1).
       y(x-1) = -1
       xy - y = -1
       xy = y - 1
       x = (y-1)/y
       Jadi, (f o f)^(-1)(x) = (x-1)/x

   4.  **(g o g)(x):**
       (g o g)(x) = g(g(x)) = g(x+3) = (x+3) + 3 = x+6
       Untuk mencari inversnya, misalkan y = x+6.
       x = y-6
       Jadi, (g o g)^(-1)(x) = x-6

b. Menghitung nilai fungsi komposisi dan inversnya:

   1.  **(f o g)(5):**
       (f o g)(5) = (5+2)/(5+3) = 7/8

   2.  **(g o f)^(-1)(5):**
       (g o f)^(-1)(5) = 1 / (4-5) = 1 / (-1) = -1

   3.  **(f o f)^(-1)(5):**
       (f o f)^(-1)(5) = (5-1)/5 = 4/5

   4.  **(g o g)^(-1)(5):**
       (g o g)^(-1)(5) = 5 - 6 = -1