February 14, 2014

 Relations: in mathematics, we refer to a group of points as a “relation” Examples: { (2, -3); (2, 5); (5,1) }

 Input: this usually refers to our “x” values. These are usually the numbers you “plug in” an equation (although you’ve seen us plug in numbers for y)  Output: this usually refers to our “y” values. This is the results we get when we plug in our input.

 Function: A special type of relation.  Functions are actually a really easy concept that has horrible definitions. Even when I try to explain them to students, I get so caught up with using the “right” language that I end up confusing students. Let’s see if I can explain it well this year!

 This is a relation (a group of points). It shows four points: {(8,-1); (9,-3); (10, -1); (13,5)}  This mapping might go with some equation where plugging in 8 gives you -1 and plugging in 9 gives you -3 and so on.

 The buttons are a little hard to read, so we are going to guess that this is what the six buttons say. Coke Diet coke Sprite Fanta Water?

 So if we press the first button, we should get a coke. If we press the second button, we should also get a coke. (two different buttons lead to coke, that is kind of normal in pop machines.) Coke Diet coke Sprite Fanta Water?

This could be our “mapping” for the pop machine. Coke Diet coke Sprite Fanta Water? Buttons Pop Coke Diet Coke Sprite Fanta Water

 This is a pop machine that I would use. It is a good Functioning machine. Buttons Pop Coke Diet Coke Sprite Fanta Water

 Now look at this machine.  Would you want to use this machine?  What if you really wanted a coke? Would you be happy with water?  This machine is probably broken. It doesn’t function properly. Buttons Pop Coke Diet Coke Sprite Fanta Water

FUNCTIONNOT A FUNCTION Buttons Pop Coke Diet Coke Sprite Fanta Water Buttons Pop Coke Diet Coke Sprite Fanta Water

 A function is a relation where each input value is paired with only one output value.  Some people sum this up by saying that “x” can’t repeat. (“y” can repeat all it wants)

 {(2,1); (-3,4); (5,2)}  {(5,8); (3,-2); (7,8)}  {(7,8); (6,9); (8,10); (7,11)}  {(1,5); (0, 5); (2, 5); (8, 4)}  {(2,-3); (4,3); (2,0)} yes no

 There are many points shown in the following graphs.  Try to think about the coordinates.  Would the “x” values have more than one “y” value attached to them?

 One fast way to tell if a graph is a function is if it “passes” the vertical line test.  If you can draw a vertical line somewhere and have it hit the graph more than once, then it is not a function.

Yes No

Yes

No

 Functions seem to work out better mathematically.  Think about what we just learned about lines.  Which of these lines is not like the others?

 Remember how writing an equation of a vertical line was difficult because we didn’t know how to plug in “undefined.”

 Functions, in general, are easier to graph, easier to calculate, and easier to plug into graphing calculators. They make sense to people because you plug in a number and only get one answer.