f.4 a maths題...function

用戶7109

2008-10-24 5:37 am

Let f(x)=ax²+bx+c,where a,b,c are real an d a does not equal to 0.

a)If α,β are the roots of f(x)-x=0, prove that α,β are also the roots of
f[f(x)]-x=0.

b)If f(x)=x²-3x+2, solve f[f(x)]-x=0.

更新1:

b part仲有2個答案 x=0 or 2 不過我識個steps

更新2:

b part仲有2個答案 x=0 or 2 不過我"唔"識個steps 識做先選做最佳

回答 (1)

魏王將張遼

2008-10-24 7:16 am

✔ 最佳答案

(a) From the given we have:
f(α) - α = 0 and f(β) - β = 0
f(α) = α and f(β) = β
So,
f[f(α)] - α = f(α) - α = 0
f[f(β)] - β = f(β) - β = 0
So α, β are the roots of f[f(x)] - x = 0
(b) Suppose α, β are the roots of f[f(x)] - x = 0
Then α, β are the also the roots of f(x) - x = 0
Therefore,
f(x) - x = x2 - 4x + 2 = 0
x = 2 + √2 or 2 - √2
Thus α, β = 2 + √2, 2 - √2

2008-11-02 11:51:30 補充：
Since 2 + √2 and 2 - √2 are roots of f[f(x)] - x = 0, f[f(x)] - x is divisible by x^2 - 4x + 2 and then:
f[f(x)] - x = (x^2 - 4x + 2)(x^2 - 2x) (By long division)
So the remaining 2 roots are 0 and 2

參考: My Maths knowledge

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