relations & functions section 2-1. definitions a relation is a description of the association...
TRANSCRIPT
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Relations & Functions
Section 2-1
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Definitions
• A relation is a description of the association between two sets of values.
• The set of input values is called the domain and the set of output values is called the range.
• For example, there is an association (hopefully!) between the color of a traffic light and the behavior of a driver approaching it.
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Traffic Light
Red
Yellow
Green
Color
Stop
Slow down
Speed up
Maintain speed
Behavior
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School Schedule
1
2
3
4
5
6
Period
Algebra 2
Gym
Chemistry
Study
Geometry
Class
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Cell Phone Direction Pad
Up
Down
Left
Right
Direction
Calculator
Inbox
Pictures
Ringtones
Action
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Multiplication
2
-1
0.5
-1.3
0
1
Number
4
-2
1
-2.6
0
2
x 2
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School Schedule (again!)
Math
Science
Gym
Study
Subject
1
2
3
4
5
6
Period
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Functions
• A function is a relation in which each input value maps to exactly one output value.
• Which of the previous examples are functions?
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Ordered Pairs
• When the input and output values are numbers, as in the multiplication example, we can think of the input and output values as x and y, and represent the relation as a collection of ordered pairs:
{(2, 4), (-1, -2), (0.5, 1), (-1.3, -2.6), (0, 0), (1, 2)}
• We can also graph these points!
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Ordered Pairs (cont’d)
x y
2 4
-1 -2
0.5 1
-1.3 -2.6
0 0
1 2
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Vertical Line Test
• When we graph a relation, we can use the vertical line test to determine whether or not it is a function.
• In order to be a function, the graph must have the property that any vertical line drawn through it only touches it once. This corresponds to each input (x) value having only one output (y) value.
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Vertical Line Test (cont’d)
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Here’s a problem…
• What if we wanted to expand the previous example to include more inputs and outputs, but following the same rule?
• We could write out some more ordered pairs:…(4, 8), (5, 10), (6, 12), (7, 14)…
… but these are just a few! There are infinitely many possible ordered pairs that we could add to that relation.
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… and a solution!
• We can represent the relation using the equation that describes the relationship between the inputs and outputs:
y = 2x
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Solution (cont’d)
• Now if want to know what output value is produced by the input value 27, we just plug 27 in for x:
y = 2(27)y = 54
• Similarly, if want to know what input value gives an output value of -13, plug -13 in for y:
-13 = 2xx = -6.5
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Example 1
• A relation is defined by the equation:y = x2 + 3
• What are some ordered pairs that are part of this relation?
(-2, 7), (-1, 4), (0, 3), (1, 4), (2, 7), (3, 12)• Is this relation a function?• Yes! For each input (x), there is exactly one
output (y) – to find it, just square and add 3!
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Example 2
• A relation is defined by the equation:|y| = x – 1
• What are some ordered pairs that are a part of the relation?
{(4, 3), (8, -7), (5, 4), (5, -4)}
• Is this relation a function?No! The input value 5 has two different output
values: 4 and -4
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Domain and Range
• Recall that the domain is the set of input values, and the range is the set of output values.
• When a relation is given as an equation, the domain and range are often difficult to figure out.
• We need to think about all the possible values of x and y in the equation.
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Example 1
y = x2 + 3• Given a number as input (x), is there always an
output value for it?• Yes – just square it and add 3.• The domain of this relation is all real numbers.
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Example 1 (cont’d)
y = x2 + 3• Given a number as output (y), can we always find
an input (x) to go with it?• No! For example, try y = -1:
-1 = x2 + 3 has no solution!• In fact, we know that x2 ≥ 0 always, so the output,
which is equal to x2 + 3, satisfies:x2 + 3 ≥ 3
• The range of this relation is {y | y ≥ 3}
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Example 2
|y| = x – 1• Given a number as input (x), is there always an
output value for it?• No! For example, try x = -5.
|y|= -5 – 1 has no solution!• In order to have a solution, we need:
x – 1 ≥ 0, or solving, x ≥ 1• The domain of this relation is {x | x ≥ 1}
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Example 2 (cont’d)
|y| = x – 1• Given a number as output (y), can we always
find an input (x) to go with it?• Yes – we can always solve for x.• The range of this relation is all real numbers.
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Function Notation
• Recall that when a relation is a function, there is exactly one output value for each input value. With functions, we sometimes use function notation to represent the output value:
f(x) = x2 + 3
Function name
Input value
Rule for finding the output value
Replaces y
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Function Notation (cont’d)
f(x) = x2 + 3• f(x) is read “f of x” and refers to the output value
when x is the input value.• f(-5) refers to the output value when 5 is used as
an input value.f(-5) = (-5)2 + 3 = 28
• f(-5) does not mean to multiply the variable f by the number -5!
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More examples
f(x) = x2 + 3
f(a) = a2 + 3
f(2a) = (2a)2 + 3 = 4a2 + 3
f(b + 5) = (b + 5)2 + 3= (b+5)(b+5) + 3 = b2 + 10b + 25 + 3
= b2 + 10b + 28