Let $f : R \to R$ be defined by $f(x) = \frac{ax^2 + ax + b}{ax + b}$. Then:

Given below are two statements:
Statement $I$: The function $f:R \rightarrow R$ defined by $f(x) = \frac{x}{1+|x|}$ is one-one.
Statement $II$: The function $f:R \rightarrow R$ defined by $f(x) = \frac{x^{2}+4x-30}{x^{2}-8x+18}$ is many-one.
In the light of the above statements, choose the correct answer from the options given below:

Let $Z$ denote the set of integers. Define $f: Z \rightarrow Z$ by $f(x) = \begin{cases} \frac{x}{2}, & x \text{ is even} \\ 0, & x \text{ is odd} \end{cases}$. Then $f$ is:

If $f: Z \rightarrow Z$ is defined by $f(x)=\begin{cases} \frac{x}{2}, & \text{if } x \text{ is even} \\ 0, & \text{if } x \text{ is odd} \end{cases}$,then $f$ is

Let $g: N \rightarrow N$ be defined as
$g(3n+1)=3n+2$
$g(3n+2)=3n+3$
$g(3n+3)=3n+1, \text{ for all } n \geq 0$
Then which of the following statements is true?

If $f: R \rightarrow R$ is defined by $f(x)=x-[x]-\frac{1}{2}$ for $x \in R$, where $[x]$ is the greatest integer not exceeding $x$, then $\{x \in R: f(x)=\frac{1}{2}\}$ is equal to :