If $f(x) = 2x$ and $g$ is the identity function,then:

If $f(x) = \cos (\log x)$,then the value of $f(x)f(4) - \frac{1}{2}\left[ f\left( \frac{x}{4} \right) + f(4x) \right]$ is:

If $f(x) = \sqrt{x^2 + x} + \frac{\tan^2 \alpha}{\sqrt{x^2 + x}}$,where $\alpha \in (0, \pi/2)$ and $x > 0$,then the value of $f(x)$ is greater than or equal to:

$f: R \rightarrow R$ is a function defined by $f(x) = \frac{1}{e^x + 2e^{-x}}$. Assertion $(A): f(c) = \frac{1}{3}$ for some values of $c \in R$. Reason $(R): 0 < f(x) \leq \frac{1}{2\sqrt{2}}$ for all $x \in R$. Then which of the following options is correct?

Let $f: X \rightarrow Y$ be a function and $A, B$ be non-void subsets of $Y$. Which of the following is true?

Let $[x]$ denote the integral part of $x \in R$. Let $g(x) = x - [x]$. Let $f(x)$ be any continuous function with $f(0) = f(1)$. Then the function $h(x) = f(g(x))$: