## Friday, January 10, 2014

### Quadratic Equations and Inequalities (Part 1)

The Gateway Arch is a 630-foot (192 m) high monument in St. Louis, in the U.S. state of Missouri. Built as a monument to the westward expansion of the United States, it is the centerpiece of the Jefferson National Expansion Memorial and has become an internationally famous symbol of St. Louis.
(http://en.wikipedia.org/wiki/Gateway_Arch)

Essential Questions
1     What makes a parabolic curve so aesthetically pleasing as an architectural structure?
2    Do you think the above structure represents a parabolic curve?

# What is a Function?

A function relates an input to an output.
 It is like a machine that has an input and an output. And the output is related somehow to the input.

 f(x) "f(x) = ... " is the classic way of writing a function. And there are other ways, as you will see!

First, it is useful to give a function a name.
The most common name is "f", but you can have other names like "g" ... or even "marmalade" if you want.
But let's use "f":
You would say "f of x equals x squared"
what goes into the function is put inside parentheses () after the name of the function:
So f(x) shows you the function is called "f", and "x" goes in
And you will often see what a function does with the input:
f(x) = x2 shows you that function "f" takes "x" and squares it.

Example: with f(x) = x2:
• an input of 4
• becomes an output of 16.
In fact we can write f(4) = 16.
Example: this tree grows 20 cm every year, so the height of the tree is related to its age using the function h:
h(age) = age × 20
So, if the age is 10 years, the height is:
h(10) = 10 × 20 = 200 cm

Here are some example values:
ageh(age) = age × 20
00
120
3.264
15300
......

## A Function is Special

But a function has special rules:
• It must work for every possible input value
• And you can only have one relationship for each input value
This can be said in one definition:

## Formal Definition of a Function

A function relates each element of a set
with exactly one element of another set
(possibly the same set).

## The Two Important Things!

 1 "...each element..." means that every element in X is related to some element in Y. We say that the function covers X (relates every element of it). (But some elements of Y might not be related to at all, which is fine.)
 2 "...exactly one..." means that a function is single valued. It will not give back 2 or more results for the same input. So "f(2) = 7 or 9" is not right!
 (one-to-many) (many-to-one) This is NOT OK in a function But this is OK in a function
If a relationship does not follow those two rules then it is not a function ... it would still be a relationship, just not a function.

### Example: The relationship x → x2

Could also be written as a table:
X: xY: x2
39
11
00
416
-416
......

It is a function, because:
• Every element in X is related to Y
• No element in X has two or more relationships
So it follows the rules.
(Notice how both 4 and -4 relate to 16, which is allowed.)

### Example: This relationship is not a function:

It is a relationship, but it is not a function, for these reasons:
• Value "3" in X has no relation in Y
• Value "4" in X has no relation in Y
• Value "5" is related to more than one value in Y
(But the fact that "6" in Y is not related to does not matter)

## Vertical Line Test

On a graph, the idea of single valued means that no vertical line would ever cross more than one value.
If it crosses more than once it is still a valid curve, but it would not be a function.

## Ordered Pairs

I said I would show you many ways to think about functions, and here is another way:
You can write the input and output of a function as an "ordered pair", such as (4,16).
They are called ordered pairs because the input always comes first, and the output second:
(input, output)
So it looks like this:
xf(x) )
Example:
(4,16) means that the function takes in "4" and gives out "16"

### Set of Ordered Pairs

A function can then be defined as a set of ordered pairs:
Example: {(2,4), (3,5), (7,3)} is a function that says
"2 is related to 4", "3 is related to 5" and "7 is related 3".
Also, notice that:
• the domain is {2,3,7} (the input values)
• and the range is {4,5,3} (the output values)

## Conclusion

• a function relates inputs to outputs
• a function takes elements from a set (the domain) and relates them to elements in a set (the codomain).
• all the outputs (the actual values related to) are together called the range
• a function is a special type of relation where:
• every element in the domain is included, and
• any input produces only one output (not this or that)
• an input and its matching output are together called an ordered pair
• so a function can also be seen as a set of ordered pairs

## Tuesday, January 7, 2014

### The Vending Machine Analogy

A function has input and output. A good analogy of this would

### Function and relation

1. If you say 2, I'll say 11. If you say 7, I'll say 26. Can you determine why I say what I say?

2. If you say 2, I'll say (-3). If you say 7, I'll say 7. Can you determine why I say what I say?

3. If you say 2, I'll say 3. If you say 7, I''ll say 21. Can you determine why I say what I say?

### Materials

1. Maths notes (given by school)
2. Maths Exercise book (for note taking and class discussion)
3. Calculator
4. Writing materials (Pens & Pencils) + Ruler
5. Learning Device
Others:
1. Textbook is optional
2. Maths file (orange)
3. Remedial (Every Monday 3 pm to 4 pm)