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 $f(x+y)=f(x)f(y)$ for all $x, y$ where $f(0) \neq 0$. If $f(5) = 2$ and $f'(0) = 3$,then $f'(5)$ is equal to

Let $f: N \rightarrow R$ be such that $f(1)=1$ and $f(1)+2 f(2)+3 f(3)+\ldots+n f(n)=n(n+1) f(n)$ for all $n \in N, n \geq 2,$ where $N$ is the set of natural numbers and $R$ is the set of real numbers. Then,the value of $f(500)$ is

Let $f: R \rightarrow R$ be defined by $f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2}$ for all $x$ and $y$. If $f^{\prime}(0)$ exists and equals $-1$ and $f(0)=1$,then $f(2)=$

If $f(x_1) - f(x_2) = f\left( \frac{x_1 - x_2}{1 - x_1 x_2} \right)$ for $x_1, x_2 \in [-1, 1]$,then $f(x)$ is

Let $f$ be a non-zero real-valued continuous function satisfying $f(x+y) = f(x) \cdot f(y)$ for all $x, y \in R$. If $f(2) = 9$,then $f(6)$ is equal to