Let $S = \{1, 2, 3, 4\}$. Then the number of elements in the set $\{f: S \times S \rightarrow S : f \text{ is onto and } f(a, b) = f(b, a) \geq a; \forall (a, b) \in S \times S\}$ is

The total number of functions $f: \{1, 2, 3, 4\} \to \{1, 2, 3, 4, 5, 6\}$ such that $f(1) + f(2) = f(3)$ is equal to:

Let $f: R \to R$ be a function. Define $g: R \to R$ by $g(x) = |f(x)|$ for all $x$. Then $g$ is

For some $a, b, c \in N$,let $f(x)=ax-3$ and $g(x)=x^b+c$,$x \in R$. If $(fog)^{-1}(x)=\left(\frac{x-7}{2}\right)^{1/3}$,then $(fog)(ac) + (gof)(b)$ is equal to $..........$

The function $f(x) = x^{\frac{1}{\ln x}}$ is:

Let $f(x) = Ax^3 - Bx - \tan x \cdot \text{sgn}(x)$ be an even function $\forall x \in R - \left\{ (2n + 1)\frac{\pi}{2}, n \in I \right\}$,where $A = \sin^2 \alpha - \sin \alpha + \frac{1}{4}$ and $B = \tan^2 \alpha + \frac{2}{\sqrt{3}} \tan \alpha + \frac{1}{3}$. Then the number of values of $\alpha$ in $\left[ -\frac{3\pi}{2}, 2\pi \right]$ is (where $\text{sgn}(x)$ denotes the signum function of $x$).