The function $$f:R \to \left[ { - {1 \over 2},{1 \over 2}} \right]$$ defined as

$$f\left( x \right) = {x \over {1 + {x^2}}}$$, is

Let $$a$$, b, c $$ \in R$$. If $$f$$(x) = ax² + bx + c is such that
$$a$$ + b + c = 3 and $$f$$(x + y) = $$f$$(x) + $$f$$(y) + xy, $$\forall x,y \in R,$$

then $$\sum\limits_{n = 1}^{10} {f(n)} $$ is equal to

For x $$ \in $$ R, x $$ \ne $$ 0, Let f₀(x) = $${1 \over {1 - x}}$$ and
f_n+1 (x) = f₀(f_n(x)), n = 0, 1, 2, . . . .

Then the value of f₁₀₀(3) + f₁$$\left( {{2 \over 3}} \right)$$ + f₂$$\left( {{3 \over 2}} \right)$$ is equal to :

If $f(x)+2 f\left(\frac{1}{x}\right)=3 x, x \neq 0$, and $\mathrm{S}=\{x \in \mathbf{R}: f(x)=f(-x)\}$; then $\mathrm{S}:$