Show that the function `f: Rvec` given by `f(x)=x^3+x` is a bijection.

Explanation

We have the function `f: R Rightarrow R` given by `f(x)=x^{3}+x`.


Injectivity :


Let `x, y in R` such that `f(x)=f(y)`


`Rightarrow x^{3}+x=y^{3}+y`


`Rightarrow x^{3}-y^{3}+x-y=0`


`Rightarrow(x-y)(x^{2}+x y+y^{2})+(x-y)=0`


`Rightarrow(x-y)(x^{2}+x y+y^{2}+1)=0`


`[ x^{2}+x y+y^{2} geq 0 text { for all } x, y in R x^{2}+x y+y^{2}+1 geq 1 text { for all } x, y in R]`


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