Notes, Lesson 1.1
Four Ways to Represent a Function
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Definitions | Identification. Is it a function? ... or not? | Function Notation |
Four Ways to Represent a Function | Check Concepts
Definitions:
Relation ANY set of ordered pairs.
Domain The set of all x-coordinates in a relation. (independent variables)
Range The set of all y-coordinates in a relation. (dependent variables)
Mapping A pictoral matching up of domain elements with their corresponding range elements.
Function A relation where each element of the domain matches up with exactly one element of the range.
Vertical Line Test A test which takes a vertical line at any horizontal position. If at any position there is a crossing at more than one point, then the relation fails the Vertical Line Test, and the relation is not a function.

More Definitions: To the Top of the Page

Coordinate Plane The plane determined by two perpendicular axes. 
Coordinate Plane

Coordinates Each point in the coordinate plane corresponds to an ordered pair of numbers called its coordinates.
X-Coordinate The first coordinate in an ordered pair. This represents the independent variable.
Y-Coordinate The second coordinate in an ordered pair. This represents the dependent variable.
Ordered Pair A set of two numbers, with the x-coordinate (independent variable) listed first, and the y-coordinate (dependent variable) listed second.

Identification. Is it a function? ... or not? To the Top of the Page
{(1,2),(2,3),(3,2)}
Function
This is a function because no x-coordinate has more than one y-coordinate matched with it.
y=3x-9
Function
This is a function because no x-coordinate will result in more than one y-coordinate.
Function
This is a function because no x-coordinate has more than one y-coordinate matched with it.
Function
The "Vertical Line Test" here shows that at no time will a vertical line ever cross the graph in more than one point. Therefore, this is a function.
{(1,2),(4,2),(6,4),(7,1),(1,3)}
Not a Function
Because of the two ordered pairs (1,2) and (1,3). There is a "violation." There is more than 1 y-coordinate for the x-coordinate 1.
Not a Function
Just one example of a "violation" is the pair of ordered pairs: (1,1), and (1,-1).
Not a Function
The two ordered pairs shown: (2,4) and (2,5) have the same x-coordinate and have different y-coordinates and therefore fail the function rule.
Not a Function
The "Vertical Line Test" here shows more than 1 point of intersection. this shows that there is at least 1 pair of points with the same x-coordinate and a different y-coordinate. This is a violation of the Vertical Line Test. This is therefore not a function.

Function Notation: To the Top of the Page
f(3)
This is read "F of three"
Given f(x)=7x-2, find f(3). Sample Problem
f(3)=19
Substitute 3 in place of x in the given function f and calculate the result.
Given g(x)=7-2x, find g(-8) Sample Problem
g(-8)=23
Substitute -8 in place of x in the given function g and calculate the result.
Given h(x)=12x+9 find h(x+3) Sample Problem
h(x+3)=12(x+3)+9
Substitute (x+3) in place of x in the given function h.
h(x+3)=12x+36+9
Distributive Property
h(x+3)=12x+45
Simplify as much as possible.

Four Ways to Represent a Function:To the Top of the Page

#
Function Description
Type of Description
1 Take the number of the step you are on, multiply it by itself,
and add 4.
Represent the function verbally, using a word description.
2
x
-4
-3
-2
-1
0
1
2
3
4
f(x)
20
13
8
5
4
5
8
13
20
Represent the function numerically, using a table of values.
3
Represent the function visually, using a graph.
4
Represent the function algebraically, using an explicit formula.



Lesson on Functions and Graphs

 
Check Concepts
Check Concepts
Check Concepts


#1: True or False: A function has only one value of x for each value of y.
   
#2: True or False: Functions pass the "vertical line test."
   
#3: For a function, all of the possible values for x are called the....
   
#4: For a function, all of the possible outcomes for y are called the...
   
#5 True or False: Functions must be continuous.

video lesson Completing the Square Flash version

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