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UNDERSTANDING FUNCTIONS
CHAPTER 1 UNDERSTANDING FUNCTIONS Lesson 1 Revisiting Functions
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R E L A T I O N S H I P
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Relation – a set of ordered pairs
(mother, child) (husband, wife) (God, man) (employer, employee) (teacher, student)
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Relation – a set of ordered pairs
{(mother, child) mother, child (husband, wife) husband, wife (God, man) God, man (employer, employee) employer, employee (teacher, student)} teacher student Domain: The set of the first coordinates of the ordered pair. { } Domain are x-values, inputs, independent variable. Range: The set of the second coordinates of the ordered pair. { , , , , } Range are y-values, outputs, dependent variable.
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These are all ways of showing a relationship between two variables.
There are many ways to represent relations: A set of ordered pairs Mapping Table of values Graph Equation These are all ways of showing a relationship between two variables. y = x+1 Equation
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TYPES OF RELATION MAPPING
{(Philippines, Manila), (Indonesia, Jakarta), (Thailand, Bangkok)} Domain Range Philippines Manila Indonesia Jakarta Thailand Bangkok
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TYPES OF RELATION MAPPING
{(Mr. Mendoza, STEM A), (Mr. Mendoza, STEM B), (Mr. Mendoza, STEM C)} Domain Range STEM A Mr. Mendoza STEM B STEM C
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TYPES OF RELATION MAPPING
{(Algebra, Mathematics), (Geometry, Mathematics), (Statistics, Mathematics)} Domain Range Algebra Geometry Mathematics Statistics
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TYPES OF RELATION MAPPING
{(Statistics, STEM), (Statistics, ABM), (Statistics, TVL), (Calculus, STEM), (Gen Math, STEM), (Gen Math, ABM), (Gen Math, TVL)} Domain Range Statistics STEM Calculus ABM Gen Math TVL
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TYPES OF RELATION One to One Many to One Many to Many Many to Many
Domain Range Domain Range Philippines Manila Algebra Indonesia Geometry Jakarta Mathematics Thailand Bangkok Statistics One to One Many to One Domain Range Domain Domain Range Range STEM A Statistics SHS STEM STEM Mr. Mendoza Calculus Calculus STEM B ABM ABM Gen Math Gen Math STEM C Tech Voc TVL One to Many Many to Many Many to Many
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Is a RELATION a FUNCTION ?
A function ( f ) is a correspondence between two sets, the domain and the range, such that for each value in the domain, there corresponds exactly one value in the range. Is a RELATION a FUNCTION ? One to One If the set of ordered pairs have different x-coordinates, It is a FUNCTION! If the set of ordered pairs have same x-coordinates, It is NOT a function Many to One One to Many Many to Many
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Practice Exercises I. Determine if the following sets represent a function. 1. { ( 5, -1) , (4, -2) , (3, -3) , (2, -4) , (1, -5)} {5, 4, 3, 2, 1} Domain: Range: {-1, -2, -3, -4, -5} Type of Relation: One to One Function: YES! 2. { ( a, 1) , (b, 1) , (c, 1) , (d, 1) , (e, 1)} {a, b, c, d, e} Domain: Range: {1} Type of Relation: Many to One Function: YES! 3. { (Manila, UST) , (Manila, Quiapo Church) , (Manila, Divisoria) , (Manila, Rizal Park)} Domain: {Manila} Range: {UST, Quiapo Church, Divisoria, Rizal Park} Function: NO! Type of Relation: One to Many
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{Algebra, Geomerty, Calculus, Statistics}
x Jan Feb Mar Apr May y 31 29 30 4. 6. Domain: Domain: {Jan, Feb, Mar, Apr, May} Range: Range: {31, 29, 30} Function: Function: YES! Many to One Type of Relation: 5. Algebra {-4, -1, 0, 2, 4} Math Geometry Domain: Range: {-3, 3, -1, 2, 1} Calculus Function: YES! Statistics Type of Relation: One to One {Math} Domain: {Algebra, Geomerty, Calculus, Statistics} Range: Function: NO! Type of Relation: One to Many
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VERTICAL LINE TEST The set of points in the xy-plane is the graph of a function if and only if every vertical line intersects the graph in EXACTLY one point.
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If a vertical line passes through a graph more than once, the graph is not the graph of a function.
NOT A FUNCTION!
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FUNCTION!
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NOT A FUNCTION!
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FUNCTION!
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NOT A FUNCTION!
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NOT A FUNCTION!
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Relation: M M M-M Relation: M M M-M
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Relation: M M M-M Relation: M M M-M
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Relation: M M M-M Relation: M M M-M
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Relation: M M M-M Relation: M M M-M
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Domain and Range R: (-2, 4) D: [-3, 3) D: -3 ≤ x < 3 D: (-3, 3)
Interval Notation Set Builder Notation D: [-3, 3) D: -3 ≤ x < 3 D: (-3, 3) R: (-2, 4] R: -2 < y ≤ 4 Interval Notation Set Builder Notation Interval Notation Set Builder Notation D: [-3, 3] D: -3 ≤ x ≤ 3 D: (-3, 3] D: -3 < x ≤ 3 R: [-2, 4] R: -2 ≤ y ≤ 4 R: [-2, 4) R: -2 ≤ y < 4
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