Given $f'(x) > 0$ and $g'(x) < 0$ for all $x \in R$,then which of the following is true?

Consider two sets $A = \{ x \in \mathbb{Z} : |(| x - 3| - 3)| \leq 1 \}$ and $B = \{ x \in \mathbb{R} - \{1, 2\} : \frac{(x - 2)(x - 4)}{x - 1} \log_{e}(|x - 2|) = 0 \}$. Then the number of onto functions $f: A \rightarrow B$ is equal to:

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

If the sets $A$ and $B$ are defined as $A = \{(x, y) : y = e^x, x \in R\}$ and $B = \{(x, y) : y = x, x \in R\}$,then:

Let the function $f: R \rightarrow R$ be defined by $f(x) = \frac{\sin x}{e^{\pi x}} \frac{(x^{2023} + 2024x + 2025)}{(x^2 - x + 3)} + \frac{2}{e^{\pi x}} \frac{(x^{2023} + 2024x + 2025)}{(x^2 - x + 3)}.$ Then the number of solutions of $f(x) = 0$ in $R$ is

$f(x) = \frac{x}{\ln x}$ and $g(x) = \frac{\ln x}{x}$. Identify the $CORRECT$ statement.