Function Rules

Relations A relation is a set of ordered pairs. The first item in an ordered pair is identified as the domain. The second item in the ordered pair is identified as the range. Let's take a look at a couple of examples:

Example 1 A relation can be written in the form of a table:

Think of a Vending Machine Think of a Vending Machine. You put in 75 cents and out pops your bag of chips. Or you put in $1.00 and out pops your Hershey Bar. There is a relationship between the amount of money that you put in the machine and what comes out! This is exactly what the "Math World" is like. It's a ton of little vending machines that "swallow" an input number (domain) and pops out another number (range).

Example 2 The following is an algebraic relation that we will call b. State the domain and range. b:{(2,4) (3,6) (4,8) (5, 10)}

The domain is: {2, 3, 4, 5} (These are all the x values of the ordered pair) The range is: {4, 6, 8, 10} (These are all the y values of the ordered pair)

The domain contains the independent variable and the range contains the dependent variable. This means that the value of the range depends on the domain. Think about the vending machine: What comes out of the machine (range) depends on what you put in (domain). You can't put in a nickel and expect a chocolate bar to pop out!

Functions! Functions are a special type of relation. In a function, each input (x coordinate) may be paired with only ONE output (y coordinate).

Let’s look at our relation from before: How can we tell if it’s a function? There are actually two ways to determine if a relation is a function. One way is to analyze the ordered pairs The other way is to use the vertical line test.

Example 1 Let's analyze our ordered pairs first. Since each input has a different output, this can be classified as a function.

b:{(2,4) (3,6) (4,8) (5, 10)} Let's verify it with the vertical line test. The vertical line test is used when you graph the ordered pairs. You imagine a vertical line being drawn through the graph. If the vertical line only touches the graph at one point, then it is a function. If the vertical line touches in more than one point, then it is NOT a function.

Example 2 You try! Is this a function? s:{(-3, 2) (-1, 6) (1,2)}

Example 3 One more! Is this a function? c:{(3,3) (-1,0) (3,-3)}