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

Let $[x]$ represent the greatest integer less than or equal to $x$,${x} = x - [x]$,$\sqrt{2} = 1.414$ and $\sqrt{3} = 1.732$. If $f(x) = \{x + [\frac{x}{1+x^2}]\}$ is a real-valued function,then $f(\sqrt{2}) + f(-\sqrt{3}) = $

Suppose $f:[-2, 2] \to R$ is defined by $f(x) = \begin{cases} -1 & \text{for } -2 \le x \le 0 \\ x - 1 & \text{for } 0 < x \le 2 \end{cases}$,then find the set $\{ x \in (-2, 2) : x \le 0 \text{ and } f(|x|) = x \}$.

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

If $a+\alpha=1, b+\beta=2$ and $af(x)+\alpha f\left(\frac{1}{x}\right)=bx+\frac{\beta}{x}$ for $x \neq 0$,then the value of the expression $\frac{f(x)+f\left(\frac{1}{x}\right)}{x+\frac{1}{x}}$ is ..... .

Let $f(x) = \frac{x^2 - 4}{x^2 + 4}$ for $|x| > 2$. Then the function $f: (- \infty, -2] \cup [2, \infty) \to (-1, 1)$ is