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2.1 Relations and Functions
Algebra 2Mrs. Aguirre
Fall 2013
Objectives
• Graph a relation, state its domain and range, and determine if the relation is a function, and
• Find the values of functions for given elements of the domain.
Relations• A relation is a mapping, or pairing,
of input values with output values.• The set of input values is called the
domain.• The set of output values is called
the range.• A relation as a function provided
there is exactly one output for each input.
• It is NOT a function if at least one input has more than one output
Input Output
-3 3
1 -2
4 1
4
Identify the Domain and Range. Then tell if the relation is a function.
Domain = {-3, 1,4}Range = {3,-2,1,4}
Function?No: input 1 is mapped onto Both -2 & 1
Notice the set notation!!!
Identify the Domain and Range. Then tell if the relation is a function.
Input Output
-3 3
1 1
3 -2
4
Domain = {-3, 1,3,4}Range = {3,1,-2}
Function?Yes: each input is mappedonto exactly one output
A Relation can be represented by a set of ordered pairs of the form (x,y)
Quadrant IX>0, y>0
Quadrant IIX<0, y>0
Quadrant IIIX<0, y<0
Quadrant IVX>0, y<0
Origin (0,0)
Graphing Relations
• To graph the relation in the previous example:
• Write as ordered pairs (-3,3), (1,-2), (1,1), (4,4)
• Plot the points
(-3,3)(4,4)
(1,1)
(1,-2)
Same with the points (-3,3), (1,1), (3,1), (4,-2)
(-3,3)
(4,-2)
(1,1) (3,1)
Vertical Line Test• You can use the vertical line test to visually
determine if a relation is a function.• Slide any vertical line (pencil) across the
graph to see if any two points lie on the same vertical line.
• If there are not two points on the same vertical line then the relation is a function.
• If there are two points on the same vertical line then the relation is NOT a function
(-3,3)(4,4)
(1,1)
(1,-2)
Use the vertical line test to visually check if the relation is a function.
Function?No, Two points are on The same vertical line.
(-3,3)
(4,-2)
(1,1) (3,1)
Use the vertical line test to visually check if the relation is a function.
Function?Yes, no two points are on the same vertical line
Graphing and Evaluating Functions
• Many functions can be represented by an equation in 2 variables: y=2x-7
• An ordered pair is a solution if the equation is true when the values of x & y are substituted into the equation.
• Ex: (2,-3) is a solution of y=2x-7 because:• -3 = 2(2) – 7• -3 = 4 – 7• -3 = -3
• In an equation, the input variable is called the independent variable.
• The output variable is called the dependent variable and depends on the value of the input variable.
• In y=2x-7 ….. X is the independent var. Y is the dependant var.
• The graph of an equation in 2 variables is the collection of all points (x,y) whose coordinates are solutions of the equation.
Graphing an equation in 2 variables
1. Construct a table of values
2. Graph enough solutions to recognize a pattern
3. Connect the points with a line or curve
Graph: y = x + 1Step 1
Table of values
Step2:Step 3:
Function Notation • By naming the function ‘f’ you can write
the function notation:
• f(x) = mx + b• “the value of f at x”
• “f of x”
• f(x) is another name for y (grown up name)
• You can use other letters for f, like g or h
Decide if the function is linear. Then evaluate for x = -2
• f(x) = -x2 – 3x + 5• Not linear….• f(-2) = -(-2)2 – 3(-2) + 5• f(-2) = 7
• g(x) = 2x + 6• Is linear because x is to the first power• g(-2) = 2(-2) + 6• g(-2) = 2• The domain for both is…..• All reals