Function notation is the way in which a function is written to precisely convey information. You may be accustomed to seeing functions written in such a way that y is written as the output of the function and is set equal to some input x.
Functions can also be written in the form of f(x), pronounced "f of x." When someone says "y is a function of x," it means that the value of y is determined by the value of x. Here, y is the dependent variable and x is the independent variable. f(x) is just the shortened form of "function of x." If you were to write the above information in the form of an expression, it would look something like:
|function of x
|function of x
Essentially, y is replaced with f(x). In f(x), the f is the name used to identify the given function, while the x is the argument of the function, and describes the input value of the function. The argument of the function must be the same as the variable used on the right side of the equation.
The reason that we replace y is because it doesn't give us enough information while f(x) gives us information about the argument of the function and at the same time identifies itself as the dependent variable.
As a comparison between notations, consider:
|y = x2 + 2
|f(x) = x2 + 2
For the equation on the left, a person may ask "What is the value of y when x = 4?" whereas for the equation on the right, one might ask "What is f(2)?"
In this particular example, it may only seem like a couple of words are being conserved, but when dealing with multiple functions and multiple arguments, function notation can be quite useful.
Note: f(x) is the most common way to denote a function, but both the function name and the argument can be changed to any symbol you want.