2.1 Relations and Functions

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2.1 Relations and Functions

• Algebra 2
• Mrs. Spitz
• Fall 2006

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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.

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Assignment

• Pp. 55-56 6-35 all

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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

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Identify the Domain and Range. Then tell if the
relation is a function.
Input Output -3 3 1 -2
4 1 4
Domain -3, 1,4 Range 3,-2,1,4
Notice the set notation!!!
Function? No input 1 is mapped onto Both -2 1
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Identify the Domain and Range. Then tell if the
relation is a function.
Input Output -3 3 1 1
3 -2 4
Function? Yes each input is mapped onto exactly
one output
Domain -3, 1,3,4 Range 3,1,-2
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A Relation can be represented by a set of ordered
pairs of the form (x,y)
Quadrant I Xgt0, ygt0
Quadrant II Xlt0, ygt0
Origin (0,0)
Quadrant IV Xgt0, ylt0
Quadrant III Xlt0, ylt0
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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

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(4,4)
(-3,3)
(1,1)
(1,-2)
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Same with the points (-3,3), (1,1), (3,1), (4,-2)
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(-3,3)
(1,1)
(3,1)
(4,-2)
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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

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Use the vertical line test to visually check if
the relation is a function.
(4,4)
(-3,3)
(1,1)
(1,-2)
Function? No, Two points are on The same
vertical line.
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Use the vertical line test to visually check if
the relation is a function.
(-3,3)
(1,1)
(3,1)
(4,-2)
Function? Yes, no two points are on the same
vertical line
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Graphing and Evaluating Functions

• Many functions can be represented by an equation
  in 2 variables y2x-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 y2x-7 because
• -3 2(2) 7
• -3 4 7
• -3 -3

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• 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 y2x-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.

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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

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Graph y x 1
Step 3
Step2
Step 1 Table of values
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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

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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