Definition of function, domain, range, onto, and 1:1.

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RELATION
Definition: a list of ordered pairs
Different ways of representing relations:
(a) As a set: {(3, 4), (­3, 9), (2, 6)} (d) Equation: y = 2x + 1
(b) As a mapping: (e) Graph:
­3 4
2 6
3 9
(c) As a table.
x y
­3 9
2 6
3 4
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DOMAIN & RANGE
Domain: input values, "x"
independent variable
Range: output values, "y"
dependent variable
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(1) Write the indicated relation ﴾2﴿ Write the relation as a
as a set of ordered pairs. set of ordered pairs.
Find domain and range Find domain &range.
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2 2
4 3
5 4
5
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{(2,2),(2,4),(2,6),(4,4),(5,5)} {(­4, 4), (­2, 4), (3, 2),
(­3, ­2), (2, ­4)}
Domain: {2, 4, 5}
Range: (1, 2, 3, 4, 5, 6} Domain: {­4, ­3, ­2, 2, 3}
Range: {­4, ­2, 2, 4}
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FUNCTIONS
Definition: a relation in which each x only
goes with one y.
Are the following relations a function? Justify your response.
(1) {(1, 2), (3, 4), (5, 6)} (2) {(0,0), (1,1), (4,2), (1,­1)}
yes
no : x=1 goes with >1 y
(3) (4)
no, (1,2) and (1,6) make it yes, it's a function.
not a function. no x goes to >1 y.
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Vertical Line Test: a graph is a function if any vertical line
only intersects it once.
a graph is not a function if there is a
vertical line that intersects it more than
once.
(5) (6)
Yes.
No.
(7) y = x2 + 7x + 6 (8) x = 3
yes, parabola no, fails the vertical
line test.
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no yes yes
no no yes yes
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A Function
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Not a Function
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Not 1 : 1
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Not Onto
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ONTO and ONE­TO­ONE
Onto = all the elements in the range are
used by the function.
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One­to­One or 1:1 = any y goes with only 1 x.
**Horizontal Line Test
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Summary:
• Domain = x = input
• Range = y = output
• Function = every x has only one y
• Onto = every y has an x
• 1:1 = every y has only one x
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