Let $S = \{1, 2, 3, 4, 5, 6\}$. Then the probability that a randomly chosen onto function $g: S \to S$ satisfies $g(3) = 2g(1)$ is:

Find the number of all one-one functions from set $A = \{1, 2, 3\}$ to itself.

The set $A$ has $4$ elements and the set $B$ has $5$ elements. Then,the number of injective mappings that can be defined from $A$ to $B$ is:

Match the following:

$(A)$ $f: R \rightarrow R$ is such that $f(x)=px+q$ $(p \neq 0)$,$\forall x \in R$	$I.$ $f$ is neither one-one nor onto
$(B)$ $f: R \rightarrow R^{+} \cup\{0\}$ is such that $f(x)=x^2$,$\forall x \in R$	$II.$ $f$ is both one-one and onto
$(C)$ $f: N \rightarrow N$ is such that $f(n)=n^2+2n+3$,$\forall n \in N$	$III.$ $f$ is one-one but not onto
$(D)$ $f: R \rightarrow R$ is such that $f(x)=2(\cos ^2 5x+\sin ^2 5x)$ $\forall x \in R$	$IV.$ $f$ is onto but not one-one
	$V.$ $f$ is a constant function and also a bijection

Let $f: R \rightarrow R$ be defined by $f(x) = \begin{cases} x + 2, & x \leq -1 \\ x^2, & -1 < x < 1 \\ 2 - x, & x \geq 1 \end{cases}$. Then the value of $f(-1.75) + f(0.5) + f(1.5)$ is

Let $A$ and $B$ be non-empty sets in $\mathbb{R}$ and $f : A \to B$ be a bijective function.
Statement $1$ : $f$ is an onto function.
Statement $2$ : There exists a function $g : B \to A$ such that $f \circ g = I_B$.