Let $$f(x)$$ be a function such that $$f(x+y)=f(x).f(y)$$ for all $$x,y\in \mathbb{N}$$. If $$f(1)=3$$ and $$\sum\limits_{k = 1}^n {f(k) = 3279} $$, then the value of n is

If $$f(x) = {{{2^{2x}}} \over {{2^{2x}} + 2}},x \in \mathbb{R}$$, then $$f\left( {{1 \over {2023}}} \right) + f\left( {{2 \over {2023}}} \right)\, + \,...\, + \,f\left( {{{2022} \over {2023}}} \right)$$ is equal to

$$ \text { Let } f(x)=a x^{2}+b x+c \text { be such that } f(1)=3, f(-2)=\lambda \text { and } $$ $$f(3)=4$$. If $$f(0)+f(1)+f(-2)+f(3)=14$$, then $$\lambda$$ is equal to :

Let $$\alpha, \beta$$ and $$\gamma$$ be three positive real numbers. Let $$f(x)=\alpha x^{5}+\beta x^{3}+\gamma x, x \in \mathbf{R}$$ and $$g: \mathbf{R} \rightarrow \mathbf{R}$$ be such that $$g(f(x))=x$$ for all $$x \in \mathbf{R}$$. If $$\mathrm{a}_{1}, \mathrm{a}_{2}, \mathrm{a}_{3}, \ldots, \mathrm{a}_{\mathrm{n}}$$ be in arithmetic progression with mean zero, then the value of $$f\left(g\left(\frac{1}{\mathrm{n}} \sum\limits_{i=1}^{\mathrm{n}} f\left(\mathrm{a}_{i}\right)\right)\right)$$ is equal to :