Let $f: \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = 2x + |x|$,then $f(2x) + f(-x) - f(x) = $

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

Function $f(x) = \sin x + \tan x + \operatorname{sgn}(x^2 - 6x + 10)$ is (where $\operatorname{sgn}$ is the signum function):

If $f(x) = \cos (\log x)$,then $f(x^2)f(y^2) - \frac{1}{2}\left[ f\left( \frac{x^2}{y^2} \right) + f(x^2y^2) \right]$ has the value

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

Let $f(x) = \max(\sin x, \cos x)$ and $g(x) = \min(\cos x, \sin x)$. Define $h(y) = f(x)y^2 + ay + g(x)$. If the equation $h(y) = 0$ has real roots for all $x \in R$,find the complete set of values of $a$.