A real-valued function $f$ satisfies the relation $f(x)f(y) = f(2xy + 3) + 3f(x + y) - 3f(y) + 6y$, for all real numbers $x$ and $y$, then the value of $f(8)$ is

The set of all possible values of $f(x)$ for which $(81)^{x}+(81)^{f(x)}=3$ is

Suppose that a real-valued function $f(x)$ of real numbers satisfies $f(x + xy) = f(x) + f(xy$) for all real $x, y,$ and that $f(2020) = 1$. Compute $f(2021)$.

Let $A=\{1,2,3\}$ and $B=\{a, b\}$. Assuming all relations from set $A$ to set $B$ are equally likely, what is the probability that a relation from $A$ to $B$ is also a function?

A set of all possible values the function $f(x)=\dfrac{x}{|x|}$, where $x \neq 0$, takes is

If a function $f(a) = max (a, 0)$ then the smallest integer value of $x$ for which the equation $f(x - 3) + 2f(x + 1) = 8$ holds true is _______.

The function $f(x) = \dfrac{x^3 - 5x^2 - 8x}{3}$ is