FUNCTIONSFebruary 14, 2014

VOCABULARY

Relations: in mathematics, we refer to a group of points as a “relation” Examples:

{ (2, -3); (2, 5); (5,1) }

VOCABULARY

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.

VOCABULARY

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!

FIRST: MAPPING POINTS

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.

NOW THINK ABOUT A POP MACHINE

The buttons are a little hard to read, so we are going to guess that this is what the six buttons say.

Coke

Coke

Diet coke

Sprite

Fanta

Water?

NOW THINK ABOUT A POP MACHINE

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

Coke

Diet coke

Sprite

Fanta

Water?

NOW THINK ABOUT A POP MACHINE

This could be our “mapping” for the pop machine.

Coke

Coke

Diet coke

Sprite

Fanta

Water?

Buttons Pop

123456

CokeDiet CokeSpriteFantaWater

THINKING OF POP MACHINES

This is a pop machine that I would use. It is a good Functioning machine.

Buttons Pop

123456

CokeDiet CokeSpriteFantaWater

THINKING OF POP MACHINES

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

123456

CokeDiet CokeSpriteFantaWater

COMPARING MACHINES

FUNCTION NOT A FUNCTION

Buttons Pop

123456

CokeDiet CokeSpriteFantaWater

Buttons Pop

123456

CokeDiet CokeSpriteFantaWater

SO NOW THE DREADED DEFINITION

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)

QUIZ: IS IT A FUNCTION?

{(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)}

yesyes

yesno

no

HOW CAN WE TELL A GRAPH IS A FUNCTION?

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?

VERTICAL LINE TEST

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.

QUIZ: IS IT A FUNCTION

YesNo

QUIZ: IS IT A FUNCTION

Yes Yes

QUIZ: IS IT A FUNCTION

Yes No

QUIZ: IS IT A FUNCTION

No No

WHY DO WE CARE ABOUT FUNCTIONS? Functions seem to work out better

mathematically. Think about what we just learned

about lines. Which of these lines is not like the

others?

WHY DO WE CARE ABOUT FUNCTIONS? Remember how writing an equation of a vertical

line was difficult because we didn’t know how to

plug in “undefined.”

WHY DO WE CARE ABOUT FUNCTIONS?

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.