functions relation such that each element x in a set a maps to exactly one element y in a set b set...
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FunctionsRelation such that each element x in a set A maps to EXACTLY ONE element y in a set B Set A: Domain = set of all numbers for which
the formula makes sense and defines real numbers (x values) x = independent variable
Set B: Range = f(x) = set of all possible f(x) as x varies through the domain (y values) y = dependent variable
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Example Find the domain of . [Think: What can’t happen?] You can’t have zero on the bottom of a
fraction.
Domain You may just write:
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Example Find the domain of .
−+¿
−Domain
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Vertical Line Test A curve in the xy-plane is the graph of a
function of x if and only if NO vertical line intersects the curve more than once
If any vertical line hits more than one point, the graph is not a function
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Example Determine if each is a function.
Yes No
Homework: p. 14 #1, 3, 11, 15, 29
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Even v. Odd Functions
Even – symmetric with respect to y – axis
(equal y values)
Odd – symmetric with respect to origin
(opposite y values)
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Function Guide
1. Constant Ex:
Even2. Identity
Odd
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3. Absolute Value Ex: Even
4. Square Root Ex: Neither
X Y
0 0
1 1
2 1.4
3 1.7
4 2
X Y
-2 2
-1 1
0 0
1 1
2 2
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5. Quadratic Ex: Even
6. Cubic Ex: Odd
X Y
-2 4
-1 1
0 0
1 1
2 4
X Y
-2 -8
-1 -1
0 0
1 1
2 8
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Composition of Functions
Ex:
In this case,
𝑓 (𝑥−3)¿ (𝑥−3)2=𝑥2−6 𝑥+9𝑔 (𝑥2) ¿ 𝑥2−3
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Piecewise Defined Functions
Greatest Integer Function
greatest integer
D = all real #’s R = all Z
X Y
0 0
0.5 0
0.9 0
1 1
“Step Function”
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Examples
Homework: p. 14 #9, 31, 41-47 odd
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Graph and Turn In for 20 Points