Fungsi f:R->R dan g:R->R ditentukan oleh f(x)=2x+1 dan

a. (gof)(-1)=-1, (gof)(2)=23, (gof)(0)=-1, (gof)(-2)=7, (gof)(3)=47. b. (gof)(x) = 4x^2 + 4x - 1, yang terbukti sesuai dengan perhitungan nilai a.

Pembahasan

Diberikan fungsi f(x) = 2x + 1 dan g(x) = x^2 - 2.

a. Mencari nilai komposisi fungsi:
(gof)(x) = g(f(x))
(gof)(x) = g(2x + 1)
(gof)(x) = (2x + 1)^2 - 2
(gof)(x) = (4x^2 + 4x + 1) - 2
(gof)(x) = 4x^2 + 4x - 1

Sekarang kita hitung nilai-nilai yang diminta:
(gof)(-1) = 4(-1)^2 + 4(-1) - 1 = 4(1) - 4 - 1 = 4 - 4 - 1 = -1
(gof)(2) = 4(2)^2 + 4(2) - 1 = 4(4) + 8 - 1 = 16 + 8 - 1 = 23
(gof)(0) = 4(0)^2 + 4(0) - 1 = 0 + 0 - 1 = -1
(gof)(-2) = 4(-2)^2 + 4(-2) - 1 = 4(4) - 8 - 1 = 16 - 8 - 1 = 7
(gof)(3) = 4(3)^2 + 4(3) - 1 = 4(9) + 12 - 1 = 36 + 12 - 1 = 47

b. Membuktikan rumus (gof)(x) dan mengecek nilai:
Kita sudah mendapatkan rumus (gof)(x) = 4x^2 + 4x - 1 pada bagian a.

Sekarang kita cek nilai-nilai dari bagian (a) menggunakan rumus ini:
(gof)(-1) = 4(-1)^2 + 4(-1) - 1 = 4(1) - 4 - 1 = -1 (Sesuai)
(gof)(2) = 4(2)^2 + 4(2) - 1 = 4(4) + 8 - 1 = 16 + 8 - 1 = 23 (Sesuai)
(gof)(0) = 4(0)^2 + 4(0) - 1 = 0 + 0 - 1 = -1 (Sesuai)
(gof)(-2) = 4(-2)^2 + 4(-2) - 1 = 4(4) - 8 - 1 = 16 - 8 - 1 = 7 (Sesuai)
(gof)(3) = 4(3)^2 + 4(3) - 1 = 4(9) + 12 - 1 = 36 + 12 - 1 = 47 (Sesuai)

Jadi, rumus (gof)(x) = 4x^2 + 4x - 1 terbukti benar dan sesuai dengan perhitungan nilai-nilai komposisi fungsi secara langsung.