// mencari fungsi invers f(x) //
f(x) = 9x²-6x+1
y = 9x²-6x+1
9x²-6x+1-y = 0
x1,2 = (6±√36-4(9)(1-y))/2(9)
= (6±√36-36(1-y))/18
= (6±6√1-(1-y))/18
= (1±√1-1+y)/3
= (1±√y)/3
x₁ = (1+√y)/3
x₂ = (1-√y)/3
// mencari g(x-2/3) //
// untuk f⁻¹(x) = x₁ //
f⁻¹o(fog)(x) = (1+√9x²+6x+1)/3
g(x) = (1+√9x²+3x+3x+1)/3
= (1+√3x(3x+1)+1(3x+1))/3
= (1+√(3x+1)²)/3
= (1+3x+1)/3
= (3x+2)/3
= x+2/3
g(x-2/3) = (x-2/3)+2/3
= x
//untuk f⁻¹(x) = x₂ //
f⁻¹o(fog)(x) = (1-√9x²+6x+1)/3
g(x) = (1-√9x²+3x+3x+1)/3
= (1-√3x(3x+1)+1(3x+1))/3
= (1-√(3x+1)²)/3
= (1-(3x+1))/3
= (-3x)/3
= -x
g(x-2/3) = -(x-2/3)
= -x+2/3
= 2/3-x
Jawabannya adalah g(x-2/3) = 2/3-x atau x
Extra:
// pembuktian //
//untuk g(x) = x+2/3 //
(fog)(x) = 9(x+2/3)²-6(x+2/3)+1
9x²+6x+1 = 9(x²+4/3x+4/9)-6x-4+1
9x²+6x+1 = 9x²+12x+4-6x-4+1
9x²+6x+1 = 9x²+6x+1
//untuk g(x) = -x //
(fog)(x) = 9(-x)²-6(-x)+1
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Verified answer
// mencari fungsi invers f(x) //
f(x) = 9x²-6x+1
y = 9x²-6x+1
9x²-6x+1-y = 0
x1,2 = (6±√36-4(9)(1-y))/2(9)
= (6±√36-36(1-y))/18
= (6±6√1-(1-y))/18
= (1±√1-1+y)/3
= (1±√y)/3
x₁ = (1+√y)/3
x₂ = (1-√y)/3
// mencari g(x-2/3) //
// untuk f⁻¹(x) = x₁ //
f⁻¹o(fog)(x) = (1+√9x²+6x+1)/3
g(x) = (1+√9x²+3x+3x+1)/3
= (1+√3x(3x+1)+1(3x+1))/3
= (1+√(3x+1)²)/3
= (1+3x+1)/3
= (3x+2)/3
= x+2/3
g(x-2/3) = (x-2/3)+2/3
= x
//untuk f⁻¹(x) = x₂ //
f⁻¹o(fog)(x) = (1-√9x²+6x+1)/3
g(x) = (1-√9x²+3x+3x+1)/3
= (1-√3x(3x+1)+1(3x+1))/3
= (1-√(3x+1)²)/3
= (1-(3x+1))/3
= (-3x)/3
= -x
g(x-2/3) = -(x-2/3)
= -x+2/3
= 2/3-x
Jawabannya adalah g(x-2/3) = 2/3-x atau x
Extra:
// pembuktian //
//untuk g(x) = x+2/3 //
(fog)(x) = 9(x+2/3)²-6(x+2/3)+1
9x²+6x+1 = 9(x²+4/3x+4/9)-6x-4+1
9x²+6x+1 = 9x²+12x+4-6x-4+1
9x²+6x+1 = 9x²+6x+1
//untuk g(x) = -x //
(fog)(x) = 9(-x)²-6(-x)+1
9x²+6x+1 = 9x²+6x+1