Functions in the form of $y = f(x)$ describe various sorts of line.
In a quadratic line, for every extra unit in $x$, then $y$ increases by roughly $2x$.
A line where for every extra unit in $x$, then $y$ doubles is exponential, $y = 2^x$.
Thease can be inversed, for example:
For every doubling of $x$, then $y$ increases by $1$ is exponential, y = $log2(x)$.
What is this called with quadratic equations?
On
These are functional equations in words:
a) $f(x+1)=f(x)+x$
b) $f(x+1)=2f(x)$
c) $f(2x)=f(x)+1$
These all have many solutions. For (a) it can be $f(x)=x(x+1)/2$ though not $x^2$; for (b) as you say $2^x$; and for (c) $\log_2(x)$. But there are many other solutions. For example for (b) it could be $17 \cos(2\pi x) 2^x$. EqWorld has a large number of examples.
If you want $f(x)=x^2$ than a possible functional equation is $f(x+1)=f(x) + 2x + 1$; another is $f(x+1)=f(x) + 2\sqrt{f(x)} + 1$.
If you want $f(x)=\sqrt{x}$ than a possible functional equation is $f(x+1)=\sqrt{f(x)^2+1}$.
Your question seems to be directed at inverse functions. Your statement "For every $1$ in $x$ then $y$ increases by $x$ is quadratic, $y = x^2$" is off by a factor of $2$, and even then it's only approximately true that $y$ increases by $2x$ when $x$ increases by $1$. The corresponding inverse statement is about the inverse function of $y=x^2$, which is $y=\sqrt{x}$: For every increase of $2y$ in $x$, $y$ increases by $1$, roughly.