If $f: R \setminus \{0\} \rightarrow R$ is such that $2 f(x) + f\left(\frac{1}{x}\right) = 4x$ and $S = \{x \in R : f(x) = f(-x)\}$,then the number of elements in $S$ is

If $f: R-\{0\} \rightarrow R$ is defined by $3 f(x)+4 f\left(\frac{1}{x}\right)=\frac{2-x}{x}$,then find the value of $f(3)$.

Let $f: R \rightarrow R$ be defined as $f(x+y)+f(x-y)=2 f(x) f(y)$ and $f\left(\frac{1}{2}\right)=-1$. Then,the value of $\sum_{k=1}^{20} \frac{1}{\sin (k) \sin (k+f(k))}$ is equal to:

$A$ real valued function $y = f(x)$ satisfies the relation $f\left( x - \frac{4}{9} \right) + 2x \le \frac{9}{4}x^2 + \frac{8}{9} \le f\left( x + \frac{4}{9} \right) - 2x$. The value of $f''(2)$ is

If $\phi (x) = a^x$,then $\{ \phi (p) \} ^3$ is equal to

If $f(f(0)) = 0$,where $f(x) = x^2 + ax + b$ and $b \neq 0$,then $a + b =$