1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

A function is like any other system. What you get out of the system depends on what you put into it. Think of the human body. We put food into our digestive system and we get something very different out. (Gross.) What we put into our bodies affects what comes out and if you don't believe that, try eating beets or asparagus.

Students should understand that functions do the exact same thing, only with numbers. (Maybe not the exact same thing.) They're all about describing relationships between two sets of numbers. These two sets of numbers must have the condition that each item from the first set of numbers pairs with exactly one item from the second set of numbers.

In other words, for every input, there is exactly one output. That's like the functions' motto.

Students should know that functions can be expressed as a pair of input and output values. A relation is a set of pairs of input and output values, usually represented in ordered pairs. For instance, the ordered pair (1, 2) means that for an input value of 1, we get an output value of 2. The ordered pair (2, 3) means that if we input 2, we get 3 out. Typically, we represent this as (x, y).

The domain is the set of inputs in a relation, also called the x-coordinates of an ordered pair. The range is the set of outputs in a relation, also called the y-coordinates of an ordered pair. If students have a hard time remembering which is which, tell them to think alphabetically. Since D comes before R in the alphabet, the domain has to come before the range. If that doesn't work, the acronym "DIXROY" might. (Domain, input, x; Range, output, y.)

To start with, students can represent functions as several ordered pairs in the form of a table. One column will be the input, or x values, and the other will be the output, or y values. For instance, we can rewrite the three points of a function (-2, 3), (0, 4), and (1, -3) in the following table.

Here, we have our clearly defined domain (D: x = -2, 0, 1) and range (R: y = 3, 4, -3). Any table with the same x value resulting in multiple y values is not a function. Remember the functions' motto?

When domains and ranges cover more than just a few select points, they're often included in parenthesis or brackets. Parenthesis indicate that the point on that end is not included, while brackets indicate that it is included. When the ∞ symbol is used, we use parentheses. Makes sense, since infinity isn't really a number and can't actually be reached.

As useful as tables are, many functions have domains and ranges that extend to positive and negative infinity. When the students' data tables start getting longer than their arms, we recommend switching to graphs. Spare a headache and a few trees, while they're at it.

A graph is a visual representation of relations. We plot the input values as x and the output values as y and treat the ordered pairs (x, y) as points on the coordinate plane. For the function above, we could plot the points as (-2, 3), (0, 4), and (1, -3).

Students should also know that functions can be represented by curves on the coordinate plane as y = f(x) where f(x) is some function of x. These are basically a bunch of points that are so close together that they form a continuous curve. For instance, these points could be part of a larger function shown by the graph below.

If students aren't sure whether they're looking at a function or not, they should perform the vertical line test: if they draw a vertical line on a graph of a relation and it intersects with the curve more than once, the relation is not a function.

The key concept to remember is that functions are systems in which one input corresponds to one output. Just like the human body is a system in which every meal corresponds to one trip to the bathroom. Or something like that.