If $f(x)$ is a quadratic function such that $f(x) f\left(\frac{1}{x}\right) = f(x) + f\left(\frac{1}{x}\right)$,then $\sqrt{f\left(\frac{2}{3}\right) + f\left(\frac{3}{2}\right)} = $

Let $a, b, c \in \mathbb{R}$. If $f(x) = ax^2 + bx + c$ is such that $a + b + c = 3$ and $f(x + y) = f(x) + f(y) + xy, \forall x, y \in \mathbb{R}$,then $\sum_{n=1}^{10} f(n)$ is equal to:

If $\phi (x) = a^x$,then $\{ \phi (p) \} ^3$ is equal to

Let $f = \{(1, 1), (2, 3), (0, -1), (-1, -3)\}$ be a function from $\mathbb{Z}$ to $\mathbb{Z}$ defined by $f(x) = ax + b$ for some integers $a$ and $b$. Determine the values of $a$ and $b$.

Let $R$ denote the set of all real numbers. Let $a_i, b_i \in R$ for $i \in \{1, 2, 3\}$. Define the functions $f: R \rightarrow R$,$g: R \rightarrow R$,and $h: R \rightarrow R$ by $f(x) = a_1 + 10x + a_2x^2 + a_3x^3 + x^4$ and $g(x) = b_1 + 3x + b_2x^2 + b_3x^3 + x^4$. Let $h(x) = f(x+1) - g(x+2)$. If $f(x) \neq g(x)$ for every $x \in R$,then the coefficient of $x^3$ in $h(x)$ is:

If $f(a) = a^2 + a + 1$,then the number of solutions of the equation $f(a^2) = 3f(a)$ is: