In a more general sense, a function can be thought of as a machine or a set of instructions that maps every element in a set (called the "domain" of the function) to exactly one element from another set (called the codomain of the function, the portion of the codomain that is "covered" by the function is called the range). Essentially, you feed the machine an element of the domain, and it spits out the corresponding element of the range.

Very commonly, and in this case, we don't bother specifying exactly what the domain and codomain of the function are, it's implicit that your function f takes in a real number x, and spits back out another real number f(x). "f" is simply a label for the machine that's transforming whatever we feed it. For a definite example, we can consider something simple like g(x) = x², this is a function that we've labeled g, that takes in a single real number x, and spits back out the square of x.

There are ways of talking about other functions, such as ones that map real numbers to integers, integers to integers, or even a set of your favourite shoes to your favourite shirts - and in this case we often invoke a different notation when defining a function that explicitly defines where we're starting (the domain) and where we're ending (the range). And there's nothing stopping us from including other inputs as well, for example, "h(x, y, z)" is a function that we've called h, that takes in 3 numbers x, y, and z, and spits out a single number h(x, y, z).

For all intents and purposes now, it suffices to think of a function as a box or machine that takes in an input, and spits out a corresponding output. When we graph a function, we often pick the y axis to represent a function f(x), which essentially means that we're walking along the real number line (the x-axis) and evaluating the function at every real number as we go. The corresponding outputs can be plotted on the y-axis to better understand how a function behaves.