Let $f(x) = x^2, x \in R$. For any $A \subseteq R$,define $g(A) = \{x \in R : f(x) \in A\}$. If $S = [0, 4]$,then which one of the following statements is not true?

Let $f:[0,1] \rightarrow \mathbb{R}$ be an injective continuous function that satisfies the condition $-1 < f(0) < f(1) < 1$. Then,the number of functions $g:[-1,1] \rightarrow [0,1]$ such that $(g \circ f)(x) = x$ for all $x \in [0,1]$ is

If $N$ denotes the set of all positive integers and if $f: N \rightarrow N$ is defined by $f(n) = \text{the sum of positive divisors of } n$, then $f(2^k \cdot 3)$, where $k$ is a positive integer, is

Let $f'(x) > 0$ and $g'(x) < 0$ for all $x \in R$. Then which of the following is true?

Consider the following statements:
$(i)$ $A$ relation is a special case of a function.
$(ii)$ $A$ function is a special case of a relation.
$(iii)$ Both relation and function are the same.

If $f(x) = \cos(\log x)$,then the value of $f(x) \cdot f(y) - \frac{1}{2} \left( f\left(\frac{x}{y}\right) + f(xy) \right)$ is: