Let $N$ denotes the set of all non-negative integers and $Z$ denotes the set of all integers. The function $f:Z\to N$ given by $f\left(x\right)=\left|x\right|$ is

Consider the following statements: 

Statement $\mathrm{I}$: The function $f:R\to R$ such that $f\left(x\right)=x^3$ for all $x\in R$ is one-one.

Statement $\mathrm{II}$: $f\left(a\right)=f\left(b\right)\Rightarrow a=b$ for all $a, b\in R$, if the function $f$ is one-one.

Which one of the following is correct in respect of the above statements?