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

If $f(x)$ and $g(x)$ are functions satisfying $f(g(x))$ = $x^3 + 3x^2 + 3x + 4$  $f(x)$ = $log^3x + 3$, then slope of the tangent to the curve $y = g(x)$ at $x =  \ -1$ is 

Let $f :R \to R$ be defined by $f(x)\,\, = \,\,\frac{x}{{1 + {x^2}}},\,x\, \in \,R.$ Then the range of $f$ is

Let $A=\{1,2,3,5,8,9\}$. Then the number of possible functions $f : A \rightarrow A$ such that $f(m \cdot n)=f(m) \cdot f(n)$ for every $m, n \in A$ with $m \cdot n \in A$ is equal to $...............$.

If $f$ is a function satisfying $f(x+y)=f(x) f(y)$ for all $x, y \in N$ such that $f(1)=3$ and $\sum\limits_{x = 1}^n {f\left( x \right) = 120,} $ find the value of $n$

Tho damnin of tho finction $\cos ^{-1}\left(\frac{2 \sin ^{-1}\left(\frac{1}{4 x^{2}-1}\right)}{\pi}\right)$ is