Equations are usually not functions (see below). An equation is a logical statement, and as such it is either true or false. Specifically, it is a statement which asserts that two mathematical expressions denote objects or quantities which are equal: equation is the situation of of making two equal. The equation has a close relative, the inequality, which is a statement that one quantity is greater than or less than another.
An expression, or formula, however can be a function, if we make it clear which variables in that formula are to serve as parameters. For instance, the quadratic formula $x^2 + 2x + 1$ will serve us as a function of $x$. There is one variable in it, so if it is going to be a function, that variable is it.
If we are confronted with a formula with multiple symbols, like $ax^2 + bx + c$ then it is not clear how it is intended to be used as a function. Some of the symbols might be considered to be held constant. Usually something which looks like $ax^2 + bx + c$ is a function of $x$, and the $a, b, c$ are coefficients, but that is not necessarily so; we are prejudiced toward interpreting it that way because it resembles a common convention.
It could easily be a function of three variables $f(a, b, c) = ax^2 + bx + c$, where $x$ is a constant, or a free variable which parametrizes the function.
This is why the function notation is useful: it makes it clear which symbols are bound as parameters and which are free. But in some situations we can drop the notation, where it is clear which variables are which, and just work with formulas.
Can equations be functions? Since an equation is itself an expression made up of symbols and operators, and has a value (boolean: true or false), it can be regarded as a function, where some or all of the symbols are considered independent variables.
This is in fact what happens when we do not know whether or not the equation is true. When we solve an equation, we find combinations of variables which make the equation true. Other combinations of variables exist which make it false.
For instance, take the equation $x^2 + y^2 = r^2$. Let us hold $r$ constant. We can regard the equation to be a boolean function: $f(x,y) := x^2 + y^2 = r^2$. Note that I used a different equal sign $:=$ to denote definition, because we now have an equal sign that can appear in a function body.
This function $f(x,y)$ is true whenever $x$ and $y$ lie on the circle of radius $r$ around the origin, and false for all other values.
The visualization software GrafEq takes this perspective on equations in order to plot their graphs.
When we consider a two-variable equation or inequality to be a boolean function, we can imagine that black ink is plotted in the 2D space for those coordinates where the inequality or equation is true, and white paper remains everywhere.
Now what if we can find an inequality which, when plotted this way, reproduces an image of its own notation? Such a formula is known is known as Tupper's self-referential formula, a form of Quine.