show f(x)=x^3+1 is injective

2007-09-19 3:16 am
show f(x)=x^3+1 is injective
更新1:

我諗樓下既解法有d問題 ====================================== eg. show f(x)=x^2 is injective let f(a)=f(b) a^2=b^2 a=b f is injective let a =1 , b=-1 a=/=b but f(a)=f(b) there's a contradiction.............. ========================================

更新2:

the correct one is : a^2-b^2=0 a+b=0 or a-b =0 a=-b or a=b therefore , f is not injective..........

回答 (3)

2007-09-19 5:52 am
參考: My Maths knowledge
2007-09-19 5:00 am
let x not=y
then f(y)-f(x)
=y^3-x^3
=(y-x)(x^2+xy+y^2)
suppose f(y)-f(x)=0
then y-x=0(rejected) or (x^2+xy+y^2)=0
then xy=-(x^2+y^2) less than0
but 0=(x^2+xy+y^2)=(x^2+2xy+y^2)-xy
=(x-y)^2-xy
so,xy=(x-y)^2 larger than 0
(remember,x not=y)
so, it has contradiction.
so,f(x)-f(y) not=0
so,f(x) not=f(y)
so,f is 1-1
proved
參考: me
2007-09-19 3:23 am
let f(x) = f(y)
x^3 + 1 = y^3 + 1
x^3 = y^3
x = y
so, when f(x) = f(y) imply x = y
it means that f(x) = x^3+1 is injective


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