Question 950516
A function is this thing that will take an input and chew it up and, based on what the input is, will give you a given output. Think of a sausage maker. You put the meat and herbs and spices on one end and it grinds the meat and produce a sausage at the other end.
So,a function is a special relationship where each input has a single output. 

It is often written just the way you have it, "f(x)" where x is the input value. But it's not limited to x. You could have S(n) or any other symbol.

Example: f(x) = x/2 ("f of x is x divided by 2") is a function, because each input "x" has a single output "x/2". We can substitute x with values:
• f(2) = 1
• f(16) = 8
• f(−10) = −5
Example: f(x) = x^2
• f(2) = 4
• f(5) = 25

But a function has special rules:

    - It must work for every possible input value
    - And it has only one relationship for each input value

A more formal definition is: A function relates each element of a set
with exactly one element of another set, which can possibly the same set.
Each element means that every element in X is related to some element in Y. We say that the function covers X, which means that it relates every element of x. Note: That only applies to the elements of x. Some elements of Y might not be related to at all, and that's fine.
"...exactly one element of another set..." means that a function is single valued. It will not give back 2 or more results for the same input.

Now think back to the sausage maker. It takes all the ingredients and produces one sausage. Likewise with functions, many-to-one is allowed but one-to-many is not.
Well, I've run out of time. I hope I've helped in some small way. Don't give up part of the difficulty with math is, it's like learning a foreign language. Once you learn the vocabulary, the rest is like putting together a jigsaw puzzle, it's very mechanical. Have faith in yourself, if you don't, who will? Best of luck