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4-3 Relations
ObjectivesStudents will be able to:
1) Represent relations as sets of ordered pairs, tables, mappings, and graphs
2) Find the inverse of a relation
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Terminology
• Relation: set of ordered pairs
• Domain: set of all x values in a relation
• Range: set of all y values in a relation
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Ways to Represent a Relation1) As a set of ordered pairs
Example: {(1, 2), (-2, 4), (0, -3)}
2) As a table
3) As a graph
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Example 1: Express each relation as a table, a graph, and a mapping. Then determine the domain and range.a) {(4, 3), (-2, -1), (-3, 3), (2, -4)}
Domain: Range:
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You try.b) {(3, 2), (5, 2), (3, -1), (0, 1)}
Domain: Range:
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Inverse: relation obtained from switching the coordinates of each ordered pair of the original relation
For example, if a relation is {(2, 1), (3, -5), (0,1)}, its inverse would be {(1, 2), (-5, 3), (1, 0)}.
Try and find the inverse of this relation. List the inverse as a set of ordered pairs.
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4-6 Functions
ObjectivesStudents will be able to:
1) Determine whether a relation is a function2) Find functional values
Note: You cannot spell function without “fun”
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Functions
• A function is a special type of relation in which each element of the domain is paired with exactly one element of the range.
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• Let’s talk about what this means by looking at a real-life example of a relation. Let’s say that our domain is students, and our range is television shows. We can create a mapping of the relation.
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• Let’s now recap using mathematical examples:
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Example 2: Determine if the relation is a function.a) {(1, 3), (2, 3), (-1, 1)}yesb) {(1, 4), (2, 1), (1, 5)}No; the x value of 1 repeatsTry c) {(3, 1), (3, 2), (3, 4)} d) {(1, -1), (2, -1)}No
yes
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Vertical Line Test
• When given a graph of a relation, one can perform a vertical line test to determine whether a relation is a function.
• If you drop in vertical lines, and they do not intersect the graph in more than one point, then the relation is a function. If they do intersect the graph in more than one point, then the relation is not a function.
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Vertical Line Test
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Example 3: Use the vertical line test to determine if the relation is a function.
a) b)
Not a functionYes, is a function
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c) d)
yesno
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Try these:e) f)
Not a functionYes, is a function
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A further look at domain and range
• Remember that a domain is the set of our x values, and a range is the set of our y values.
• We can also determine the domain and range for linear equations, quadratic equations, absolute value equations, all types of equations.
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Example 4: For each graph, determine the domain and range.a) Domain:
Range:
b) Domain:
Range:
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c) Domain:
Range:
d) Domain:
Range:
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Try these.e) f)
Domain: Domain:Range: Range:
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Function Notation• Sometimes, an equation like might be written
as . This is what is referred to as function notation.
• The notation indicates that an equation is a function. In other words, if you graphed the equation, it would pass the vertical line test.
• Functions can be evaluated by taking the value in parenthesis and substituting each variable in the function with that value. After substitution is complete, simplify and combine any like terms.
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a) f(-2) b) g(5) c) f(2d)
Try these.d) g(-4) e) f(3p) f) g(2a)