If $f(x)$ is a polynomial function satisfying the condition $f(x) \cdot f(1/x) = f(x) + f(1/x)$ and $f(2) = 9$,then:

If $f(x)$ is a function such that $f(x+y)=f(x)+f(y)$ and $f(1)=7$, then $\sum_{r=1}^n f(r)=$

Let $f(x)$ be a differentiable function which satisfies the equation $f(xy) = f(x) + f(y)$ for all $x > 0, y > 0$. Then $f'(x)$ is equal to:

Let a function $f : R \rightarrow R$ be defined such that $3f(2x^2 - 3x + 5) + 2f(3x^2 - 2x + 4) = x^2 - 7x + 9$ for all $x \in R$. Then the value of $f(5)$ is:

Let $f: R \rightarrow R$ be such that $f$ is injective and $f(x)f(y) = f(x+y)$ for all $x, y \in R$. If $f(x), f(y),$ and $f(z)$ are in $GP$, then $x, y,$ and $z$ are in:

Let $f : N \rightarrow R$ be a function such that $f(x+y)=2 f(x) f(y)$ for natural numbers $x$ and $y$. If $f(1)=2$,then the value of $\alpha$ for which $\sum_{k=1}^{10} f(\alpha+k)=\frac{512}{3}(2^{20}-1)$ holds,is