Let $\left(f_n\right)_{n\in\mathbb{N}}$ be a sequence of functions, where $f_n:\mathbb{R}\to\mathbb{R}$ and be $g:\mathbb{R}^2 \to \mathbb{R}$ some function. I know that if I plug in a sequence $\left(x_n\right)_{n\in\mathbb{N}}$ into $g$ I can construct a sequence of functions, $\left(g_n\right)_{n\in\mathbb{N}}$, defined by $g_n:=g(x_n,y)$.
I am wondering if a sequence of functions, $\left(f_n\right)_{n\in\mathbb{N}}$, can be interpreted as a function of two variables?
In this case $F:\mathbb{N}\times \mathbb{R} \to \mathbb{R}$, where $F(n,x)=f_n(x)$. I am a little bit unsure because sometimes we have sequences of functions where $n$ doesn't appear in the expression of a member $f_n(x)$ but it changes the domain of $f_n$, e.g. $$f_n(x) = \begin{cases} x &, x \leqslant \frac{1}{n}\\ 2 - x &, \frac{1}{n} < x \leqslant \frac{2}{n}\\ 0 &, x > \frac{2}{n} .\end{cases}$$
In your example, n does not change the domain of the function.
If the domain of $f_n$ is $\mathbb{R}$ $\forall n$ then the domain of F is $\mathbb{N} \times \mathbb{R}$ and F can be interpreted as you require. If the domain of $f_0$ differs from that of $f_1$, for example, then F is only defined for values of x that are present in both domains.