What is a function and how does it differ from a polynomial?

A function is a notion somewhat abstract to the non-experienced reader, but can be explained as follows:

You have a domain - that is a set, e.g. the set of all yellow birds. You also have a magic box with no bottom - that is your function. On putting a yellow bird in the box, exactly one thing comes from the bottom of the box; that could be an airplane, or a number, so imagine the box being very, very big. In most cases in the function, which we'll call "f" for convenience, we put only real numbers, say 12.1345. The result that the function gives us is f(12.1345), which as we saw above can easily be an Airbus, or a Concorde, but for the purposes of high school mathematics, f(12.1345) would be a real number. E.g.  f(12.1345) = pi   (which is approximately 3.1416).

We can denote functions as f(x)=x+1 for all real values of x, or we can denote them in a more general manner as:

f is such that:

1) f(x) = x if x is positive

2) f(0) = 12

3) f(x) = x^2 if x is negative.

Note that in this example for any real value of x we can determine f(x), so that our domain is the set of all real numbers.

The first example (f(x)=x+1 for all real values of x) is an example of a polynomial function, whereas the second example is not a polynomial, but rather three polynomials stitched together: 

1) x, the identity polynomial; 

2) 12, a constant polynomial;

3) x^2, a second degree polynomial;

Any polynomial is a function, and the main difference between polynomials and functions is that polynomials are very well-behaved. That is if you take p to be a polynomial, then p(12) and p(12.00000001) would most likely be two very close numbers. Also, polynomials p(x) can be written as linear expressions of 1, x, x^2, x^3,... (that is, multiply some of these by constants and add them); e.g. p(x)=3+14x-135x^3+x^2005.

On the other hand, there is a function f for which f(12) is a black hole, and f(12.00000001) is a mice - two not very close objects; a more high school - acceptable example is f(12)=1000 and f(12.00000001)=0, where f(x) could be "defined" (could exist) for x=12 and x=12.00000001 only (it is a matter of choice, really).