Let $f: R \rightarrow R$ be defined by $f(x) = x^{2} - \frac{x^{2}}{1+x^{2}}$ for all $x \in R$. Then,

Consider the function $f(x) = x^3 - 8x^2 + 20x - 13$. The function $f: R \rightarrow R$ is:

Let $X$ be a set with exactly $5$ elements and $Y$ be a set with exactly $7$ elements. If $\alpha$ is the number of one-one functions from $X$ to $Y$ and $\beta$ is the number of onto functions from $Y$ to $X$,then the value of $\frac{1}{5!}(\beta-\alpha)$ is.

If $f: \{1, 2, 3, 4\} \to \{1, 2, 3, 4\}$ is a function such that $|f(\alpha) - \alpha| \leqslant 1$ for all $\alpha \in \{1, 2, 3, 4\}$,then the total number of such functions is:

If $f: N \rightarrow R$ is defined by $f(1)=-1$ and $f(n+1)=3f(n)+2$ for $n \geq 1$, then $f$ is

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: