Question

Diketahui fungsi f(x) = 2x2 + 3, g(x) = 3x – 1, dan h(x) = x3.
Tentukanlah:

Answer 1

Jawaban:

a.

(f.g)(x) = f(g(x)) = f(3x-1) = 2(3x-1)² + 3 = 18x² - 12x + 5

(f.h)(x) = f(h(x)) = f(x³) = 2x³² + 3 = 2x⁶ + 3

(g.f)(x) = g(f(x)) = g(2x² + 3) = 6x² + 8

(g.h)(x) = g(h(x)) = g(x³) = 3x³ - 1

(h.g)(x) = h(g(x)) = h(3x-1) = (3x-1)³ = 27x³ - 27x² + 9x - 1

(h.f)(x) = h(f(x)) = h(2x² + 3) = (2x² + 3)³ = 8x⁶ + 36x⁴ + 54x² + 27

b.

(f.g)-1(x) = f-1(g-1(x)) = f-1((x+1)/3) = √((x+1)/6)

(f.h)-1(x) = f-1(h-1(x)) = f-1(x^(1/3)) = √((x-3)/4)

(g.f)-1(x) = g-1(f-1(x)) = g-1(√((x-3)/2)) = (1/3)√(2x-6)+1

(g.h)-1(x) = g-1(h-1(x)) = g-1(x^(1/3)) = (x+1)/3

(h.g)-1(x) = h-1(g-1(x)) = h-1((x+1)/3) = ((x+1)/27)^(1/3)

(h.f)-1(x) = h-1(f-1(x)) = h-1(√((x-3)/2)) = ((x-3)/8)^(1/3)

c.

(f.g.h)(x) = f(g(h(x))) = f(g(x³)) = f(3x³ - 1) = 2(3x³ - 1)² + 3 = 54x⁶ - 36x³ + 5

(f.h.g)(x) = f(h(g(x))) = f((3x-1)³) = 2(3x-1)⁶ + 3 = 54x⁶ - 162x⁵ + 189x⁴ - 126x³ + 41x² - 6x + 3

(g.f.h)(x) = g(f(h(x))) = g(2x⁶ + 3) = 6x⁴ + 8

(g.h.f)(x) = g(h(f(x))) = g((2x² + 3)³) = 27x² - 9

(h.g.f)(x) = h(g(f(x))) = h(6x² + 8) = (6x² + 8)³ = 216x⁶ + 432x⁴ + 288x² + 64

(h.f.g)(x) = h(f(g(x))) = h(3x-1)² + 3 = (3x-1)⁶

d.

(f.g.h)-1(x) = h-1(g-1(f-1(x))) = h-1(g-1(√((x-3)/2))) = h-1((((x-3)/2)+1)/3) = ((x+5)^(1/6))^3

(f.h.g)-1(x) = g-1(h-1(f-1(x))) = g-1(h-1(√((x-3)/6))) = g-1((∛(x-3))^3) = ∛(x-3)

(g.f.h)-1(x) = h-1(f-1(g-1(x))) = h-1(f-1((x-8)/6)) = h-1(√((x-8)/2)) = ((x-8)/4)^(1/3)

(g.h.f)-1(x) = f-1(h-1(g-1(x))) = f-1(h-1((x+1)/3)) = f-1(((((x+1)/3)^(1/3))^2+3)/2) = (2((((x+1)/3)^(1/3))^2+3))-3

(h.g.f)-1(x) = f-1(g-1(h-1(x))) = f-1(g-1(∛(x))) = f-1((∛(x)+1)/3) = ((3(√x-1))/2)^(1/3)

(h.f.g)-1(x) = g-1(f-1(h-1(x))) = g-1(f-1((3x-1)^(1/3))) = g-1(√(((3x-1)^(1/3)-3)/2)) = ((3(x+1))^(1/6))/√2