Functions

Math 123

September 24-26, 2008

Another boring topic?

• Functions are one of the mathematical concepts that students understand the least.

• Why do we study functions?

• How would you define a function?

My favorite definition

• I like to think of a function as a machine, a “black box.” The input goes into the box, which performs with its machinery whatever complicated operations necessary, and then “spits out” the result of its computations, the output. Each time you put a particular number in the machine, it is always going to give you the same answer.

• You know something is not a function if there are two outputs for an input.

• Functions can be represented verbally, with formulas, tables, graphs, arrow diagrams, or with ordered pairs. The first four are the most common.

• You know something is not a function is the graph representing that function fails the so-called vertical line test.

• A function is not a formula, table, or graph.• It is not always possible to find a formula for a

function.

Notation/terminology • Domain: all possible inputs.• Codomain: the set that the outputs are elements of. • Range: all possible outputs. This is not the same as

codomain: the range is a subset of the codomain. • If the domain of f is A and codomain is B, then we write f:

A B and the result of applying f to x in A is denoted f(x).• If a function is defined by the rule f(x) = 2x, where x is any

whole number, then the domain of this function is the set of whole numbers, the codomain is also the set of whole numbers, but the range is the set of even numbers.

• After this section, I promise we will stay away from theory for a while, and instead play more with manipulatives and think about how children learn mathematics.