If $f(x) = \frac{x - |x|}{|x|}$,then $f(-1) = $

The number of roots of the equation $\sqrt{2}+e^{\cosh^{-1} x}-e^{\sinh^{-1} x}=0$ is

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

Let $P(x)$ be a polynomial with real coefficients such that $P(\sin^2 x) = P(\cos^2 x)$ for all $x \in [0, \pi/2)$. Consider the following statements:
$I.$ $P(x)$ is an even function.
$II.$ $P(x)$ can be expressed as a polynomial in $(2x - 1)^2$.
$III.$ $P(x)$ is a polynomial of even degree.
Then,

Let $f(x) = x^{2}$ and $g(x) = 2x + 1$ be two real functions. Find $(f+g)(x)$,$(f-g)(x)$,$(fg)(x)$,and $(\frac{f}{g})(x)$.

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: