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Functions Basic Properties

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7/9/2013 Functions 2 Dow Jones Closing Average Daily closing stock prices for a week Functions in Everyday Life Question: What is the formula for the function ? Does this graph represent a function ? M T W Th F Stock Price in Dollars

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7/9/2013 Functions 3 3 Average daily high temperature Historical daily data points for July Functions in Everyday Life Question: What is the formula for the function ? Does this graph represent a function ? Daily High (F°)

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7/9/2013 Functions 4 4 PC Color Palette Table lookup Functions in Everyday Life Question: What is the graph ? Does this table represent a function ? red orn yel chr grn..... blu dkb mag … …. Index Color

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7/9/2013 Functions 5 5 Relational Database Table Student data functionally related to id Functions in Everyday Life Id No Last Name First Name 1396 Miller James 1372 Burrows Susan 1448 Wilson Dorothy 1531 Bronson Charles KEY ATTRIBUTES Question: What is the graph ? Does this table represent a function ?

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7/9/2013 Functions 6 6 Functions in Everyday Life Function with formula and graph Travel time as a function of speed t = d r r t Question: Is the function the graph … or the formula … or something else ?

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7/9/2013 Functions 7 Functions in Everyday Life Function with formula and graph Conversion of Fahrenheit to Centigrade () C = 5 9 F – 32 F C Question: Is the function the graph … or the formula … or something else ?

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7/9/2013 Functions 8 8 Functions in Everyday Life Lets Review Functions with graphs but no formula DOW Jones Closing Average Average daily temperatures Functions with no graph, no formula PC Color Palette Student relational database table

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7/9/2013 Functions 9 9 Functions in Everyday Life Lets Review Functions with a formula and a graph Travel time based on rate of travel Conversion of Fahrenheit to Centigrade Question: How do we define all of these … in one simple way ?

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7/9/2013 Functions 10 Functions in Everyday Life Function Characterization What is the common characteristic ? Ordered Pairs: (day, average), (day, temperature F ), (student id, name), (speed, travel time), (F, C ) Relates one set of data with another

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7/9/2013 Functions 11 Ordered Pairs Ordered Pair Composed of two components: First Component and Second Component ( a, b )( a, b ) First Component Second Component Component Types: Can be any kind of objects

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7/9/2013 Functions 12 Ordered Pairs Sets of Ordered Pairs Notation: Example: Colors Ordered Pair: { ( a, b ), ( c, d ), ( e, f ), ( g, h ) } Set of ordered pairs: { ( orange, blue ), ( red, green ), … } ( red, green )

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7/9/2013 Functions 13 Ordered Pairs Example: Numbers Ordered pair: Set of ordered pairs: { ( -6, 10 ), ( 3, 0 ), ( 2, 7 ), ( -2, 0 ) } ( 3, 0 ) Question: Does the order of the pairs matter ?

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7/9/2013 Functions 14 Ordered Pairs Numbers and Colors Ordered pair: Set of ordered pairs: { ( 0, black ), ( 1, red ), ( 2, yellow ) … } ( 3, blue ) Example: Question: Does this appear to be a function ?

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7/9/2013 Functions 15 Ordered Pairs Historic Figures Ordered pair: Set of ordered pairs: { ( 0123, Jim Bowie ), ( 0124, Anson Jones ) … } ( 0129, Sam Houston ) Example: Question: How many entries can this set have ?

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7/9/2013 Functions 16 { …,, … } Functions Functional Relationships Ordered pairs are the key Set A Set B a b (, ) a b a b S = Question: Is S a relation ? YES Is S a function ?Maybe … it depends

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7/9/2013 Functions 17 { } Functions Function Relates each member of the domain with exactly one member of the range domain range a b c d (, ) a b S = Question: Is S a function ?YES… probably j k, … (, ) j k, c d,

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7/9/2013 Functions 18 domain range Functions Function Maps each domain element to exactly one range element a b c d Question: Is S a function ? j k Is ( c, b ) OK ? YES ! Are ( c, b ) and ( c, k ) OK ? NO ! Not with ( c, b ) and ( c, k ) !! { } (, ) a b S =, … (, ) c k, c b,

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7/9/2013 Functions 19 Functions Function We know what a function does … … but what is a function ? Definition: A function is a set of ordered pairs, no two of which have the same first component.

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7/9/2013 Functions 20 Functions Function Alternate Definition (Textbook): A function is a relation in which each domain element is related to exactly one range element.

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7/9/2013 Functions 21 Whats A Function? Class Definition of Function A function is a set of ordered pairs, no two of which have the same first component. Note : The definition is a complete sentence The definition names the thing it defines: function

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7/9/2013 Functions 22 Functions Examples 1. A = { (b, d), (g, h), (x, y) } 2. B = { (y, a), (y, b), (y, c), (y, d) } 3. C = { (a, y), (b, y), (c, y), (d, y) } 4. D = { (1, 3), (3, 1), (1, 1), (3, 3) } Which of these are functions ? Which are not ? 1 and 3 2 and 4 WHY ?

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7/9/2013 Functions 23 Describing Functions Functions as finite sets – list notation Examples A = { ( a, b ), ( c, d ), ( e, f ), ( g, h ) } B = { ( -6, 10 ), ( 3, 0 ), ( 2, 7 ), ( -2, 0 ) } C = { } ( org, blu ), ( red, grn ), ( blk, brn ) Question: Any two pairs with same first component?

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7/9/2013 Functions 24 Describing Functions Functions as large sets Large sets too big to list Example: – set builder F is the set of all ordered pairs (x,y) such that y = 2x for all real x F = { } (x, y) | y = 2x, x is any real number – maybe infinite

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7/9/2013 Functions 25 Describing Functions Set builder examples G = { (x, y) y = 2x, x > 0 } S = { (x, y) x = student-ID, y a student } D = { (d, c) d = day, c = DOW Jones close } Do all functions have formulas? Do functions always have graphs? Question:

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7/9/2013 Functions 26 Describing Functions Do functions always have graphs? No, but ordered pairs make graphing easy F = { ( -5, 4 ), ( -2, 1 ), ( 2, 7 ), ( 4, 1 ) } a b (-5, 4) (-2, 1) (2, 7) (4, 1) What is the graph ? … and now? Scatterplot Line graph Question:

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7/9/2013 Functions 27 y Function Graphs f = { (x, y) | y = 2x – 1 } x (-1, -3) (1, 1) (2, 3) (3, 5) (4, 7) The Graph of f

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7/9/2013 Functions 28 Domain = [ 0, 4 ] Finding Domain and Range Consider: f = { (x, y) | y = 2x – 1, 0 x 4 } x y (0, -1) (4, 7) The Graph of f Range = [ -1, 7 ] Note: For THIS function, the domain and range are closed intervals = { x | 0 x 4 } = { y | -1 y 7}

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7/9/2013 Functions 29 The Language of Functions Notation f = { (x, y) y = 2x, x is any real number } f is the name of the function f ( x ) is the value of the function at x { (x, y) y = 2x, x is any real number } is the function itself

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7/9/2013 Functions 30 The Language of Functions Notation f = { (x, y) y = 2x, x is any real number } x is the independent variable (or input) y is the dependent variable (or output) Input determines output We say y is a function of x … OR … output is a function of input

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7/9/2013 Functions 31 Functions as Tables Example Tuition at John Q. Public Junior College is $250 per semester hour for total hours less than 12 For 12 or more hours the tuition is a flat charge of $2800 For hours H and tuition T construct a table for a variety of course loads up to 18 hours T H

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7/9/2013 Functions 32 Functions as Tables Example T H Question: Is H a function of T ? Is T a function of H ? Does T determine H ? Does H determine T ? H is NOT a function of T T is IS a function of H Question: Does this last fit the definition ? No Yes So …

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7/9/2013 Functions 33 Graphs, Sets and Tables Ordered Pairs and Functions Input Output x y (x, y) Ordered Pair Input Output x y f = { y = 2x + 1 } (x, y) (x, y) Table

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7/9/2013 Functions 34 Graph and Functions Vertical Line Test No vertical line touches the graph of a function at more than one point Function ? YES Function? NO Function? YES (x 1, y 1 ) (x 1, y 2 ) x = x 1

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7/9/2013 Functions 35 Graph and Functions Vertical Line Test Examples Does any vertical line cut the graph more than once ? Function? YES Function? NO Function? YES x y x y x y y = x 2 x = y 2 x = y 2, y > 0

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7/9/2013 Functions 36 Relations Domain Range a b m c k j h Relation S = { ( a, m ), ( b, m ), ( c, m ), ( c, h ), … }

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7/9/2013 Functions 37 Set of all second components Set of all first components Relation As A Set { } ( a, b ), ( c, d ), ( e, f ), ( g, h ) Domain: Range: { a, c, e, g } { b, d, f, h } NOTE: DOMAIN and RANGE are SETS of objects Relation: Any set of ordered pairs Notaton:

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7/9/2013 Functions 38 Examples Relations A = { ( a, b ), ( c, d ), ( c, f ), ( g, b ) } Domain of A = Range of A = { a, c, g } { b, d, f }

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7/9/2013 Functions 39 Relations B = { ( -6, 7 ), ( 3, 0 ), ( 2, 7 ), ( -2, 0 ) } Domain of B = Range of B = { -6, -2, 2, 3 } Examples { 0, 7 }

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7/9/2013 Functions 40 Examples Relations S = { (1493, Bob), (3872, Sally), (2840, Bob), (5492, Mary) } Domain of S = Range of S = { 1493, 3872, 2840, 5492 } { Bob, Sally, Mary }

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7/9/2013 Functions 41 The Word RANGE The two meanings of RANGE Measure of dispersion of 1-variable data Maximum-minimum difference A real number, a statistic Target for a relation Derived from ordered pairs Set of all second components

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7/9/2013 Functions 42 Graphical Presentations One-variable data in one dimension Example: -21, -13, -8, 3, 5, 9, 12, A plot of numbers

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7/9/2013 Functions 43 Graphical Presentations Two-variable data in two dimensions Example: x y (-7,2) (-5,-3) (-1,4) (2,-4) (5,-2) (8,6) x y A plot of ordered pairs

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7/9/2013 Functions 44 Think about it !

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