If $e^{f(x)}=\frac{10+x}{10-x}, x \in(-10,10)$ and $f(x)=k f\left(\frac{200 x}{100+x^2}\right)$, then $k$ is equal to

If $f(x) = \frac{2^x}{2^x + \sqrt{2}}$,$x \in R$,then $\sum_{k=1}^{81} f\left(\frac{k}{82}\right)$ is equal to :

Let $f(x)=x^2+2x+2$,$g(x)=-x^2+2x-1$,and $a, b$ be the extreme values of $f(x)$ and $g(x)$ respectively. If $c$ is the extreme value of $\frac{f}{g}(x)$ (for $x \neq 1$),then $a+2b+5c+4=$

Let $S = \{1, 2, 3, 4, 5, 6\}$ and $X$ be the set of all relations $R$ from $S$ to $S$ that satisfy both the following properties:
$i$. $R$ has exactly $6$ elements.
$ii$. For each $(a, b) \in R$,we have $|a-b| \geq 2$.
Let $Y = \{R \in X : \text{The range of } R \text{ has exactly one element}\}$ and $Z = \{R \in X : R \text{ is a function from } S \text{ to } S\}$.
Let $n(A)$ denote the number of elements in a set $A$.
$(1)$ If $n(X) = {}^{m}C_{6}$,then the value of $m$ is. . . .
$(2)$ If the value of $n(Y) + n(Z)$ is $k^{2}$,then $|k|$ is. . . .

If $X$ and $Y$ are two non-empty sets where $f: X \to Y$ is a function defined such that $f(C) = \{f(x) : x \in C\}$ for $C \subseteq X$ and $f^{-1}(D) = \{x : f(x) \in D\}$ for $D \subseteq Y$,then for any $A \subseteq X$ and $B \subseteq Y$,which of the following is true?

Let $f : R \to R$ be a function defined by $f(x) = - \frac{|x|^3 + |x|}{1 + x^2}$; then the graph of $f(x)$ lies in the :-