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

As follows:

圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/Sep07/Injective.jpg

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

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