For the mapping $f: R-\{1\} \rightarrow R-\{2\}$,given by $f(x)=\frac{2x}{x-1}$,which of the following is correct?

$f: N \rightarrow N, f(x) = x^3$ is . . . . . . .

If the function $f: R \rightarrow R$ is defined by $f(x) = |x|(x - \sin x)$,then which of the following statements is $TRUE$?

Let the functions $f$ and $g$ be $f: [0, \frac{\pi}{2}] \rightarrow R$ given by $f(x) = \sin x$ and $g: [0, \frac{\pi}{2}] \rightarrow R$ given by $g(x) = \cos x$,where $R$ is the set of real numbers. Consider the following statements:
Statement $(I)$: $f$ and $g$ are one-one.
Statement $(II)$: $f+g$ is one-one.
Which of the following is correct?

$A$ mapping from $\mathbb{N}$ to $\mathbb{N}$ is defined as follows: $f: \mathbb{N} \rightarrow \mathbb{N}$ where $f(n) = (n+5)^2$ for all $n \in \mathbb{N}$ (where $\mathbb{N}$ is the set of natural numbers). Then:

The function $f: N \rightarrow Z$ defined by $f(n) = \begin{cases} \frac{n}{2} & , n \text{ is even} \\ -\left(\frac{n-1}{2}\right) & , n \text{ is odd} \end{cases}$ is . . . . . . .