Let $\phi (x) = (x) + {2^{\log _x^3}} - {3^{\log _x^2}}$ then

Let $A=\{1,2,3,5,8,9\}$. Then the number of possible functions $f : A \rightarrow A$ such that $f(m \cdot n)=f(m) \cdot f(n)$ for every $m, n \in A$ with $m \cdot n \in A$ is equal to $...............$.

If the graph of non-constant function is symmetric about the point $(3,4)$ , then the value of $\sum\limits_{r = 0}^6 {f(r) + f(3)} $ is equal to

The domain of the function $f(x) = \frac{{{{\sin }^{ - 1}}(3 - x)}}{{\ln (|x|\; - 2)}}$ is

The range of function $f : R \rightarrow  R$, $f(x) = \frac{{{{(x\, + \,1)}^4}}}{{{x^4} + \,1}}$ is

Let $f: R \rightarrow R$ be a continuous function such that $f\left(x^2\right)=f\left(x^3\right)$ for all $x \in R$. Consider the following statements.

$I.$ $f$ is an odd function.

$II.$ $f$ is an even function.

$III$. $f$ is differentiable everywhere. Then,