If $f:R \to R$ satisfies $f(x + y) = f(x) + f(y)$, for all $x,\;y \in R$ and $f(1) = 7$, then $\sum\limits_{r = 1}^n {f(r)} $ is

The domain of definition of the function $y(x)$ given by ${2^x} + {2^y} = 2$ is

Let ${f_k}\left( x \right) = \frac{1}{k}\left( {{{\sin }^k}x + {{\cos }^k}x} \right)\;,x \in R$ and $k \ge 1$, then ${f_4}\left( x \right) – {f_6}\left( x \right)$ is equal to

Let $f(x) = {\cos ^{ – 1}}\left( {\frac{{2x}}{{1 + {x^2}}}} \right) + {\sin ^{ – 1}}\left( {\frac{{1 – {x^2}}}{{1 + {x^2}}}} \right)$ then the value of $f(1) + f(2)$, is –

For a suitably chosen real constant $a$, let a function, $f: R-\{-a\} \rightarrow R$ be defined by $f(x)=\frac{a-x}{a+x} .$ Further suppose that for any real number $x \neq- a$ and $f( x ) \neq- a ,( fof )( x )= x .$ Then $f\left(-\frac{1}{2}\right)$ is equal to