Let $R = \{ a, b, c, d, e \}$ and $S = \{1, 2, 3, 4\}$. The total number of onto functions $f: R \rightarrow S$ such that $f(a) \neq 1$ is equal to $.............$.

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 $f \circ g = I_B$.

Let $E = \{ 1, 2, 3, 4 \} $ and $F = \{ 1, 2 \} $. Then the number of onto functions from $E$ to $F$ is

The function $f: R \rightarrow R$ defined by $f(x)=\frac{x}{\sqrt{1+x^2}}$ is

Show that the Signum Function $f: R \rightarrow R$,given by $f(x) = \begin{cases} 1, & \text{if } x > 0 \\ 0, & \text{if } x = 0 \\ -1, & \text{if } x < 0 \end{cases}$ is neither one-one nor onto.

If $f(x) = \begin{cases} [x] & \text{if } -3 < x \leq -1 \\ |x| & \text{if } -1 < x < 1 \\ |[x]| & \text{if } 1 \leq x < 3 \end{cases}$,then the set $\{x : f(x) \geq 0\}$ is equal to