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:

The number of solutions of the equation $|x^2 - 2|x|| = 2^x$ is:

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

The function $y = \frac{2x - 1}{x - 2}$ $(x \neq 2)$:

Let $f$ and $g$ be increasing and decreasing functions,respectively,from $[0, \infty)$ to $[0, \infty)$. Let $h(x) = f(g(x))$. If $h(0) = 0$,then $h(x) - h(1)$ is:

The number of non-constant functions $f: X \to Y$ where $X = \{0, 1, 2\}$ and $Y = \{1, 2, 3, 4, 5, 6, 7, 8\}$ such that $f(i) \leq f(j)$ whenever $i < j$ is: