g(X)=2×+3
f(X)=....
2. F(X)=×-2
(g0f) (X)= ײ-6×+8
g (X) =....
Jawab:
- f(x) = (1/4)x² – (1/2)x + 9/4
- g(x) = x² – 2x
Penjelasan dengan langkah-langkah:
Komposisi Fungsi
Nomor 1
(f ∘ g)(x) = x² + 2x + 3
g(x) = 2x + 3
f(x) = ...?
y = g(x)
⇔ y = 2x + 3
⇔ y – 3 = 2x
⇔ x = (y – 3)/2
(f ∘ g)(x) = x² + 2x + 3
⇔ f(g(x)) = x² + 2x + 3
⇔ f(y) = [(y – 3)/2]² + 2[(y – 3)/2] + 3
⇔ f(y) = (1/4)(y² – 6y + 9) + y – 3 + 3
⇔ f(y) = (1/4)y² – (6/4)y + 9/4 + y
⇔ f(y) = (1/4)y² – (2/4)y + 9/4
⇔ f(y) = (1/4)y² – (1/2)y + 9/4
∴ f(x) = (1/4)x² – (1/2)x + 9/4
__________________________
Nomor 2
f(x) = x – 2
(g ∘ f)(x) = x² – 6x + 8
g(x) = ...?
(g ∘ f)(x) = x² – 6x + 8
⇔ g(f(x)) = x² – 6x + 8
⇔ g(x – 2) = x² – 6x + 8
..... faktorkan ruas kanan
⇔ g(x – 2) = (x – 2)(x – 4)
..... ubah bentuk x – 4
⇔ g(x – 2) = (x – 2)((x – 2) – 2)
..... ganti semua (x – 2) dengan x
⇔ g(x) = (x)(x – 2)
⇔ g(x) = x² – 2x
∴ g(x) = x² – 2x
Cara penyelesaian nomor 2 ini berbeda dengan nomor 1 di atas, karena salah satu faktor dari x² – 6x + 8 adalah x – 2.
Khusus untuk nomor 2 ini, kita lakukan verifikasi.
f(x) = x – 2 ; g(x) = x² – 2x
(g ∘ f)(x) = g(f(x))
⇔ (g ∘ f)(x) = g(x – 2)
⇔ (g ∘ f)(x) = (x – 2)² – 2(x – 2)
⇔ (g ∘ f)(x) = x² – 4x + 4 – 2x + 4
⇔ (g ∘ f)(x) = x² – 6x + 8
⇔ (benar)
[answer.2.content]
