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}) = $

Let $f(x) = -1 + |x - 2|$ and $g(x) = 1 - |x|$. Then the set of all points where $fog$ is discontinuous is

Let $P(x)$ be a polynomial with real coefficients such that $P(\sin^2 x) = P(\cos^2 x)$ for all $x \in [0, \pi/2)$. Consider the following statements:
$I.$ $P(x)$ is an even function.
$II.$ $P(x)$ can be expressed as a polynomial in $(2x - 1)^2$.
$III.$ $P(x)$ is a polynomial of even degree.
Then,

Consider two sets $A = \{ x \in \mathbb{Z} : |(| x - 3| - 3)| \leq 1 \}$ and $B = \{ x \in \mathbb{R} - \{1, 2\} : \frac{(x - 2)(x - 4)}{x - 1} \log_{e}(|x - 2|) = 0 \}$. Then the number of onto functions $f: A \rightarrow B$ is equal to:

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-$II$
$(i) 14$
$(ii) 4$
$(iii) -11$
$(iv) -1$
$(v) 1$
$(vi) 0$

Which of the following real-valued functions is/are not even functions?