Functions

Soon after you learn how to count you may find yourself playing with numbers to create trends or patterns. As an example, you might play a game called “Count by twos” where you say 2, 4, 6, 8…

We can match the numbers of the game with the counting numbers. For the following (x,y) we have x as the counting number and y as the “count by twos” number: (1,2), (2,4), (3,6), (4,8)

We have another way to write that:

For the above notation the letter is the symbol for the function. The parenthesis holds the argument. The argument is .

The equation for the function is shown on the right side of the equals sign. In most discussions we just say “function” when we are referring to the equation of the function.

If a function has only one independent variable, it can be any letter; it usually or . For a function of x, the function will be a sum of terms with each term being a scalar and to some power. An example is shown below where the set {a,b,c} is a set of numbers and we are using them as scalars:

We can do something now that normally isn’t done until you reach Tensor Calculus.

If we need dozens or hundreds of variables then the alphabet won’t supply enough for our need. We can resort to using a single letter and putting a numerical subscript on it to serve as an identifier. We will do so below in a way that seems confusing at first (we didn’t count 1,2,3) but it will make sense a few paragraphs later.

The set becomes . Our function becomes the following:

We wish to remind you of three facts that are a part of Elementary Algebra:

We can use these to modify the way we write our function:

We leave this discussion of functions after noticing that all three terms of the function share something in common, they fit the template…

.,..where is 0, 1 or 2.