If $f(x) = \cos (\log x)$,then $f(x)f(y) - \frac{1}{2}[f(x/y) + f(xy)] = $

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 the function $f(x) = x^2 + x + \sin x - \cos x + \log(1 + |x|)$ be defined over the interval $[0, 1]$. The odd extension of $f(x)$ to the interval $[-1, 1]$ is:

If $f : R \rightarrow R$ is defined by $f(x) = \begin{cases} x + 4, & x < -4 \\ 3x + 2, & -4 \leq x < 4 \\ x - 4, & x \geq 4 \end{cases}$ then the correct matching of List-$I$ from List-$II$ is :
List-$I$
$(A) f(-5) + f(-4)$
$(B) f(|f(-8)|)$
$(C) f(f(-7) + f(3))$
$(D) f(f(f(f(0)))) + 1$
List-$II$
$(i) 14$
$(ii) 4$
$(iii) -11$
$(iv) -1$
$(v) 1$
$(vi) 0$

Let $f:[0,2] \rightarrow R$ be the function defined by $f(x)=(3-\sin(2\pi x)) \sin(\pi x-\frac{\pi}{4})-\sin(3\pi x+\frac{\pi}{4})$. If $\alpha, \beta \in[0,2]$ are such that $\{x \in[0,2]: f(x) \geq 0\}=[\alpha, \beta]$,then the value of $\beta-\alpha$ is:

If $f: R \setminus \{0\} \rightarrow R$ is defined by $f(x) = x + \frac{1}{x}$,then the value of $(f(x))^2 =$