Exactly how many functions $f: \mathbb{Q} \rightarrow \mathbb{Q}$ exist such that $f(x+y) = f(x) + f(y)$ and $f(xy) = f(x)f(y)$ for all $x, y \in \mathbb{Q}$?

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

Let $f: R-\{0\} \rightarrow R$ be a function satisfying $f\left(\frac{x}{y}\right)=\frac{f(x)}{f(y)}$ for all $x, y$ where $f(y) \neq 0$. If $f^{\prime}(1)=2024$,then:

$f$ is a real-valued function satisfying the relation $f\left(3x + \frac{1}{2x}\right) = 9x^2 + \frac{1}{4x^2}$. If $f\left(x + \frac{1}{x}\right) = 1$, then $x =$

Let $f: R \rightarrow R$ satisfy $f(x+y)=2^{x} f(y)+4^{y} f(x)$ for all $x, y \in R$. If $f(2)=3$,then $14 \cdot \frac{f^{\prime}(4)}{f^{\prime}(2)}$ is equal to

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: