Kisi Kisi PAT Matematika Kelas XI no 70

70. Jika (𝑓 ◦ 𝑔)(𝑥) = 2𝑥^2 + 8𝑥 + 4 dan 𝑔(𝑥) = 2𝑥 + 4, maka 𝑓⁻¹(𝑥)=⋯

A. x + 9

B. 2 +√𝑥

C. 𝑥^2 − 4𝑥 − 3

D. 2 + √(𝑥 + 1)

E. 2 + √(𝑥 + 7)

Jawaban :

𝑔(𝑥) = 2𝑥 + 4

(𝑓 ◦ 𝑔)(𝑥) = 2𝑥^2 + 8𝑥 + 4

f (g (x)) = 2𝑥^2 + 8𝑥 + 4

f (2x + 4) = 2𝑥^2 + 8𝑥 + 4

misalkan :

2x + 4 = y

2x = y - 4

x = 1/2 y - 2

sehingga, 

f (2x + 4) = 2𝑥^2 + 8𝑥 + 4

f (y) = 2(1/2 y - 2)^2 + 8(1/2 y - 2) + 4

f (y) = 2(1/4 y^2 - 2y + 4) + 4y - 16 + 4

f (y) = 1/2 y^2 - 4y + 8 + 4y - 16 + 4

f (y) = 1/2 y^2 - 4

f (x) = 1/2 x^2 - 4

mencari nilai inversnya :

f (x) = 1/2 x^2 - 4

y = 1/2 x^2 - 4

y + 4 = 1/2 x^2

2y + 8 = x^2

x = √ (2y + 8)

𝑓⁻¹(y) = √ (2y + 8)

𝑓⁻¹(x) = √ (2x + 8)