Let $f: R \rightarrow R$ be a function defined by $f(x) = \frac{2e^{2x}}{e^{2x} + e}$. Then $f\left(\frac{1}{100}\right) + f\left(\frac{2}{100}\right) + f\left(\frac{3}{100}\right) + \dots + f\left(\frac{99}{100}\right)$ is equal to

If $f: R \rightarrow R$ and $g: R \rightarrow R$ are given by $f(x)=|x|$ and $g(x)=[x]$ for each $x \in R$,then $\{x \in R: g(f(x)) \leq f(g(x))\}$ is equal to

Let $f(x) = \begin{cases} x-1, & x \text{ is even} \\ 2x, & x \text{ is odd} \end{cases}$. If for some $a \in N, f(f(f(a))) = 21$,then $\lim_{x \rightarrow a^{-}} \left\{ \frac{|x|^3}{a} - \left[ \frac{x}{a} \right] \right\}$,where $[t]$ denotes the greatest integer less than or equal to $t$,is equal to:

Let $f'(x) > 0$ and $g'(x) < 0$ for all $x \in R$. Then which of the following is true?

$A$ function $f(x)$ is given by $f(x) = \frac{5^{x}}{5^{x} + \sqrt{5}}$. Then the sum of the series $f\left(\frac{1}{20}\right) + f\left(\frac{2}{20}\right) + f\left(\frac{3}{20}\right) + \ldots + f\left(\frac{39}{20}\right)$ is equal to ....... .

The function $f(x) = \sec \left[ \log \left( x + \sqrt{1 + x^2} \right) \right]$ is . . . . . . function.