A function $f(x)$ is given by $f(x)=\frac{5^{x}}{5^{x}+5}$, then the sum of the series

$f\left(\frac{1}{20}\right)+f\left(\frac{2}{20}\right)+f\left(\frac{3}{20}\right)+\ldots \ldots+f\left(\frac{39}{20}\right)$ is equal to ....... .

If $f(x) = \cos (\log x)$, then $f(x)f(y) - \frac{1}{2}[f(x/y) + f(xy)] = $

Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by

$f(x)=\frac{2 x}{\sqrt{1+9 x^2}}$. If the composition of $f, \underbrace{(f \circ f \circ f \circ \ldots \circ f)}_{10 \text { times }}(x)=\frac{2^{10} x}{\sqrt{1+9 \alpha x^2}}$, then the value of $\sqrt{3 \alpha+1}$ is equal to....................

If $f(x) = \frac{1}{{\sqrt {x + 2\sqrt {2x - 4} } }} + \frac{1}{{\sqrt {x - 2\sqrt {2x - 4} } }}$ for $x > 2$, then $f(11) = $

If $f(x) = \frac{{{x^2} - 1}}{{{x^2} + 1}}$, for every real numbers. then the minimum value of $f$

Let $A= \{1, 2, 3, 4\}$ and $R : A \to A$ be the relation defined by $R = \{ (1, 1), (2, 3), (3, 4), ( 4, 2) \}$. The correct statement is