Question

1] Jika f(x) = 1/2(a^x + a^-x) dan g(x) = 1/2(a^x - a^-x), maka f(x) . f(y) + g(x) . g(y) sama dengan ?

2] Jika x^1/2 + x^-1/2 = 2, maka nilai dari x^2 + x^-2 adalah ?

3] Jika x^1/2 + x^-1/2 = 2, maka x^3/2 + x^-3/2 sama dengan ?

Answer 1

E|<sp0|\|en

1]

f(x) = 1/2 (a^x + a^-x)

g(x) = 1/2 (a^x - a^-x)

•

f(x) . f(y) + g(x) . g(y)

= (1/2)^(a^x + a^-x) . (1/2)^ (a^y + a^-y) + (1/2)^(a^x + a^-x) . (1/2)^ (a^y + a^-y)

= 2 . (2^-1)^((a^x + a^-x) + (a^y + a^-y))

= 2^(1 - a^x - a^-x - a^y - a^-y)

2]

→ (a + b)² = a² + b² + 2ab

•

x^1/2 + x^-1/2 = 2

(x^1/2 + x^-1/2)² = 2²

x + 1/x + 2 = 4

x + 1/x = 2

•

(x + 1/x)² = 2²

x² + 1/x² + 2 = 4

x^2 + x^-2 = 2 ✔

3]

x^1/2 + x^-1/2 = 2

•

(x + 1/x)(x² + 1/x²) = x³ + x + 1/x + 1/x³

x³ + 1/x³ = (x + 1/x)(x² + 1/x²) - (x + 1/x)

•

x³ + 1/x³

= (x + 1/x)(x² + 1/x²) - (x + 1/x)

= 2 × 2 - 2

= 2 ✔