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**Basic Properties Basic Properties**

Functions Basic Properties Functions In this section we explore the concept of function, after number, probably the most common concept in all of mathematics. In addition to conceptual views there is discussion of notation, calculation and various applications of functions. Functions 7/9/2013 1

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**Functions in Everyday Life**

Basic Properties Functions in Everyday Life Dow Jones Closing Average Daily closing stock prices for a week 8149 8700 8516 8332 M T W Th F Stock Price in Dollars Functions in Everyday Life The Dow Jones Closing Average (DJIA) of 30 “industrials” is computed and posted at the close of business each business day. It can be retrieved from a variety of web sites, including the NYSE (New York Stock Exchange). Data spanning a day, a week, a month, a quarter or a year can be retrieved and displayed graphically. The illustration shows the DOW for one week. Since there is only one data point per day (the closing average across all the DOW components), the graph is a line graph that “connects the dots” in order to make trends more visible. Does this graph represent a function? How do we know if we don’t know what a function is? That is, is it true that for each day there is one and only one value for the DOW? The answer is clearly yes, so this does represent a function of time. Question: Is there a formula for this function? If you think you have one, that knowledge can be translated into a great deal of wealth. Clearly this function has a graph, but no formula that can be used for prediction purposes. This is characteristic of all historical data. Question: Does this graph represent a function ? What is the formula for the function ? Functions 7/9/2013 Functions 7/9/2013 2

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**Functions in Everyday Life**

Basic Properties Functions in Everyday Life Average daily high temperature Historical daily data points for July 1 10 20 31 90 95 100 105 110 Daily High (F°) Functions in Everyday Life The illustration shows the daily high temperature for one month (July). Since there is only one data point per day (the high temperature), the graph is a line graph that “connects the dots” in order to make trends more visible. Does this graph represent a function? How do we know if we don’t know what a function is? That is, is it true that for each day there is one and only one value for the DOW? The answer is clearly yes, so this does represent a function of time. Question: Is there a formula that will yield the temperature, given any day of the month as input? If you think you have one and are willing to share it, you will be very popular amongst the TV weathermen. Clearly this function has a graph, but no formula that can be used for prediction. This is characteristic of all historical data. Question: Does this graph represent a function ? What is the formula for the function ? Functions 7/9/2013 3 Functions 7/9/2013 3

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**Functions in Everyday Life**

Basic Properties Functions in Everyday Life PC Color Palette Table lookup Index Color red orn yel chr grn blu dkb mag …. …. Functions in Everyday Life The PC color palette table is used by PC programmers to provide colors for display objects. The software selects an index value, say 43, which it passes through the hardware adaptation layer (HAL) to the pixel map for the desired object in the hardware. The hardware changes all the pixels in the pixel map to the selected color. Note that for each index value there is one and only one color in the table. So, the relationship represented by the table is a function giving color as a function of the index value (1, 2, 3, ... , 256). Question: Does this function have a graph? A formula? Question: Does this table represent a function ? What is the graph ? Functions 7/9/2013 4 Functions 7/9/2013 4

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**Functions in Everyday Life**

Basic Properties Functions in Everyday Life Relational Database Table Student data functionally related to id Id No Last Name First Name Miller James Burrows Susan Wilson Dorothy Bronson Charles Functions in Everyday Life The relational database table (or relation) contains rows each of which represents a student and contains three attributes of each student: student id, last name, and first name. The tenets of relational database design dictate that each row contains a unique key, the student id. This is the defining concept for a function. That is, each id value is related to exactly one row in the table, i.e. each id is related to one and only one last-name-first-name pair. This is an apparent departure from the usual ordered-pair description of function. However, it is still the same, the second component being itself an ordered pair. If we include other information, let’s say phone number, address, classification, etc. we can see that the second component is actually an ordered n-tuple. Question: Does this function have a graph? A formula? KEY ATTRIBUTES Question: Does this table represent a function ? What is the graph ? Functions 7/9/2013 5 Functions 7/9/2013 5

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**Functions in Everyday Life**

Basic Properties Functions in Everyday Life Function with formula and graph Travel time as a function of speed r t t = d r Functions in Everyday Life Using the well known relationship d = rt, where d is distance traveled, r is the rate of travel (speed), and t is the time traveled, we can find time t as a function of r by solving for t: It is clear that for each value of r there is one and only one value of t, the defining concept for a function. In this case there is both a formula and a graph for t as a function of r. Question: Is the function the graph … or the formula … or something else ? Functions 7/9/2013 6 Functions 7/9/2013 6

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**Functions in Everyday Life**

Basic Properties Functions in Everyday Life Function with formula and graph Conversion of Fahrenheit to Centigrade F C ● 100 ( ) C = 5 9 F – 32 ● 32 212 Functions in Everyday Life In this example, we wish to find Centigrade temperature C as a function of Fahrenheit temperature F. We know that water freezes at 0 C° and 32 F°, and that water boils at 100 C° and 212 F°. This is a linear relationship, so using the two points (32, 0) and (212, 100), we can find the slope of the graph at 5/9 and C-intercept –160/9. Hence we find C as a function of F, given by or more commonly Clearly each value of F produces one and only one value of C, making C a function of F. Using the formula shown above for C we can plot the graph of the function. This function has both a graph and a formula. Question: Is the function the graph … or the formula … or something else ? Functions 7/9/2013 Functions 7/9/2013 7

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**Functions in Everyday Life**

Basic Properties Functions in Everyday Life Let’s 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 Functions in Everyday Life: Review Reviewing the examples: Some functions have graphs but no formula These are normally the result of modeling historic data Some functions have no graph and no formula These generally use at least one non-numeric variable The first two items represent functional relationships that can be graphed on a variety of time scales, but have no formula – so not predictive. The second two items represent functional relationships that have no graph and no formula. These are strictly “table lookup” relationships. How do we characterize all these relationships? What do they all have in common that would allow a common characterization? Functions 7/9/2013 8 Functions 7/9/2013 8

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**Functions in Everyday Life**

Basic Properties Functions in Everyday Life Let’s Review Functions with a formula and a graph Travel time based on rate of travel Conversion of Fahrenheit to Centigrade Question: Functions in Everyday Life: Review Reviewing the examples: Some functions do have a formula and hence a graph. These functions use numeric variables. The last two items represent functional relationships that have associated formulas. These allow for predictions, the stuff of modern day computer models. The formula also allows for plotting a graph. How do we characterize this and the preceding relationships? What do they all have in common that would allow a common characterization? How do we define all of these … in one simple way ? Functions 7/9/2013 9 Functions 7/9/2013 9

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**Functions in Everyday Life**

Basic Properties 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 Function Characterization To grasp a central concept that defines all functions, let us look for some property they all have in common. What is an obvious common characteristic? All the examples shown have ordered pairs of some sort as part of their description. We note the pair examples. Let us explore the notion of ordered pairs and their relationship with what we think of as functions. One characteristic that should be clear is the a function relates data in one set (the domain) with that of another set (the range). Functions 7/9/2013 10 Functions 7/9/2013 10

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**Ordered Pairs Ordered Pair Composed of two components:**

Basic Properties Ordered Pair Composed of two components: First Component and Second Component ( a , b ) First Component Second Component The Ordered Pair The concept of function, as a mathematical object, is based on the notion of ordered pair. Here the concept is presented in a general way and the common notation for it made clear. Emphasis is placed on the generality of the notion. The components can be anything, real or imagined, and not necessarily normal mathematical objects. Component Types: Can be any kind of objects Functions 7/9/2013 11 Functions 7/9/2013 11

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**Ordered Pairs { ( a , b ) , ( c , d ) , ( e , f ) , ( g , h ) }**

Basic Properties Sets of Ordered Pairs Notation: Example: Colors Ordered Pair: { ( a , b ) , ( c , d ) , ( e , f ) , ( g , h ) } ( red , green ) The Ordered Pair The concept of function, as a mathematical object, is based on the notion of ordered pair. Here the concept is presented in a general way and the common notation for it made clear. Emphasis is placed on the generality of the notion. It does not apply only to pairs of numbers but also to pairs of any objects. Sets of Ordered Pairs Introduction of the notion of a set of ordered pairs is a prelude to the definition of function as a set of ordered pairs. This sets the stage for understanding what kind of object a function is. Set of ordered pairs: { ( orange , blue ) , ( red , green ) , … } Functions 7/9/2013 12 Functions 7/9/2013 12

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**Does the order of the pairs matter ?**

Ordered Pairs Basic Properties Example: Numbers Ordered pair: Set of ordered pairs: ( 3 , 0 ) { ( -6 , 10 ) , ( 3 , 0 ) , ( 2 , 7 ) , ( -2 , 0 ) } Examples This example pairs numbers. There is no apriori meaning associated with these numbers, though such association is certainly not ruled out. It should be noted that the only ordering involved is within each pair, first component and second component, which is why they are called ordered pairs. The ordering of the pairs within the set holds bears no inherent meaning, since sets are unordered. The salient feature here is that each first component is paired with one and only one second component; in other words, no two (distinct) ordered pairs have the same first component. Question: Does the order of the pairs matter ? Functions 7/9/2013 13 Functions 7/9/2013 13

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**Does this appear to be a function ?**

Ordered Pairs Basic Properties Example: Numbers and Colors Ordered pair: Set of ordered pairs: ( 3 , blue ) { ( 0 , black ) , ( 1 , red ) , ( 2 , yellow ) … } Examples This example shows pairing of numbers with colors might be part of a color palette table in a computer. A low level program could be coded to select a color by number from the table and pass this information to the hardware adaptation layer (HAL) in the computer hardware. Clearly, it is essential that each number be mapped to one and only one color. Question: Does this appear to be a function ? Functions 7/9/2013 14 Functions 7/9/2013 14

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**How many entries can this set have ?**

Ordered Pairs Basic Properties Example: Historic Figures Ordered pair: Set of ordered pairs: ( 0129 , Sam Houston ) { ( 0123 , Jim Bowie ), ( 0124 , Anson Jones) … } Examples This example shows pairing of numbers with other kinds of objects. The pairings of numbers with people might be part of a personnel database. This example can be a nice lead in to the notion of function by using uniquely valued first components; that is, each id identifies one and only one person. By the way, who were Sam Houston, Jim Bowie, and Anson Jones? Question: How many entries can this set have ? Functions 7/9/2013 15 Functions 7/9/2013 15

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**Functions Functional Relationships Ordered pairs are the key Set A**

Basic Properties Functions Functional Relationships Ordered pairs are the key Set A Set B a ( , ) a b b Functional Relationships The key to understanding functions is the understanding of ordered pairs. The illustration shows the pairing of elements from a set A (the domain) with elements of set B (the range) to form ordered pairs with first components from A and second components from B. We then show collecting these ordered pairs in a set we shall call S, for convenience. Since S is a set of ordered pairs, it meets the definition of relation and is therefore a relation. The question is whether or not S is a function. We can’t tell for sure from the information shown. S = { … , , … } ( , ) a b Question: Is S a relation ? YES Is S a function ? Maybe … it depends Functions 7/9/2013 16 Functions 7/9/2013 16

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Basic Properties Functions Function Relates each member of the domain with exactly one member of the range domain range a b c d j k Function We continue collecting ordered pairs from elements of A and B. By collecting ordered pairs with different first components we see that we have a possible function, certainly a relation. We emphasize that each element of the domain is mapped (related) to one and only one element of the range. The set S is a function, provided we do not collect an ordered pair with a first component that has already been used in another (distinct) ordered pair. This is a another way to think about mapping each domain element to a unique range element. { } S = ( , ) a b , ( , ) c d , ( , ) j k , … Question: Is S a function ? YES … probably Functions 7/9/2013 17 Functions 7/9/2013 17

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

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Basic Properties 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. Function: Definition Having seen how functions can be represented and how they behave, we still need to know what a function is. This requires a very careful definition to avoid confusion. The first definition expresses exactly what a function is as a set of ordered pairs. This definition holds up for all possible kinds of functions and requires only the fundamental concept of ordered pair and that of set. The second definition defines function as a relation, with the added restriction for unique mapping into the range. It does require the definitions of relation, domain and range, which is a bit more prerequisite knowledge than is required by the first definition. Functions 7/9/2013 19 Functions 7/9/2013 19

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Basic Properties Functions Function Alternate Definition (Textbook): A function is a relation in which each domain element is related to exactly one range element. Function: Alternate Definition The first definition expresses exactly what a function is as a set of ordered pairs. This definition holds up for all possible kinds of functions and requires only the fundamental concept of ordered pair and that of set. This second definition, from the textbook, defines function as a relation, with the added restriction for unique mapping into the range. It does require the definitions of relation, domain and range, which is a bit more prerequisite knowledge than is required by the first definition. Functions 7/9/2013 20 Functions 7/9/2013 20

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**What’s A Function? Class Definition of Function**

Basic Properties 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 Definition of Function The formal definition of function provides a concrete, testable statement of exactly what a function is (and by extension, what it is not). This particular definition is free of fuzzy language about such things as relations, relationships and processes, all of which easily fall into the applications of the function concept. It is also a very simple statement to learn and is based on nothing more than the ordered pair notion already introduced. Simplicity and clarity are the hallmarks of any good definition. As reminders to the student, it is worth noting that precise language here and throughout mathematics is necessary to avoid later confusion. This level of precision is generally quite new to the entry level student and should stressed, along with the reasons for it. For example, use of words such as GROUP, which have other meanings in mathematics, should be avoided. Comparisons in mathematics are generally made in binary fashion, hence the use of the phrase “no two” instead of “none” (which literally means “no one” and would not make sense in this context). The use of “repeated first component” is just fuzzy language and should be avoided for clarity. Students CAN learn the definition (if you ask it enough times on tests), which is the first step in UNDERSTANDING the concept. Functions 7/9/2013 21 Functions 7/9/2013 21

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**Functions Examples 1. A = { (b, d), (g, h), (x, y) }**

Basic Properties 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 ? Functions: Examples Here we show four relations, only two of which are functions. We note that functions don’t necessarily deal with numbers. We also see, in Example 3, a constant function with constant value y. In each of Examples 2 and 4 we can find two (distinct) ordered pairs with the same first component. In Example 2, the one element of the domain, y, is related (mapped) to every element of the range. In Example 4, we see 1 being mapped to each of 1 and 3, and 3 being mapped to each of 1 and 3 also. 1 and 3 2 and 4 WHY ? Functions 7/9/2013 22 Functions 7/9/2013 22

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**( org , blu ) , ( red , grn ) , ( blk , brn )**

Describing Functions Basic Properties 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 ) Functions As Finite Sets Having already introduced functions as sets of ordered pairs, the question of how to describe them remains. Here we see finite sets in the use of the list notation. With the examples on the screen, some basic questions are asked: Are these sets functions? Do functions always have a formula? Do functions always have a graph? Many students come to the course with the notion that the answer to these questions is YES. The examples should persuade them that this is in fact NOT the case. It is hoped the student will learn something more about the concept of function from the confrontation with preconceived notions. Question: Any two pairs with same first component? Functions 7/9/2013 23 Functions 7/9/2013 23

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**Describing Functions Functions as large sets – set builder |**

Basic Properties Functions as large sets Large sets too big to list Example: – set builder – maybe infinite F = { } (x, y) | y = 2x , x is any real number F is the set of all ordered pairs (x,y) Functions As Large Sets Having already introduced functions as sets of ordered pairs, the question of how to describe them remains. Here we see “large” sets (both finite and infinite) in the use of the list notation and the set builder notation, respectively. The animations for the function F is intended to show the student how to read the notation. It is hoped from this that the student will also learn how to write the notation. With proper encouragement this should be possible. such that y = 2x for all real x Functions 7/9/2013 24 Functions 7/9/2013 24

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**Describing Functions Set builder examples**

Basic Properties 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 } Describing Functions Here we see examples of the set builder notation. Note that the first example is for an infinite set, while the last two examples are for finite, but large, sets. With the examples on the screen, two basic questions are asked: Do functions always have formulas? Do functions always have graphs? Many students come to the course with the notion that the answer to both questions is YES. The examples should persuade them that this is in fact NOT the case. It is hoped the student will learn something more about the concept of function from the confrontation with preconceived notions. Question: Do all functions have formulas? Do functions always have graphs? Functions 7/9/2013 25 Functions 7/9/2013 25

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**Describing Functions Do functions always have graphs? **

Basic Properties Do functions always have graphs? No, but ordered pairs make graphing easy F = { ( -5 , 4 ) , ( -2 , 1 ) , ( 2 , 7 ) , ( 4 , 1 ) } a b Question: (2, 7) What is the graph ? Do Functions Always Have Graphs? While function do not always have graphs, when they do the ordered pair notation is very convenient for plotting the points on the graph. In this example we have only four ordered pairs in the function, so the corresponding graph has only four points. The coordinates of the points are of course just the components of the ordered pairs. Note here that the horizontal axis is marked a, not x, and the vertical axis is marked b, not y. This corresponds to a set of ordered pairs (a, b) and when plotted the first components become the horizontal coordinates and the second components become the vertical coordinates. Once the four points are plotted from the set of ordered pairs, we have the graph of the function – the complete graph ! Many students assume that to “finish” the graph we must “connect the dots” with line segments. To obtain a graph with lines, line segments or curves requires a set with infinitely many ordered pairs. Clearly some sort of formula will have to be included with the function to produce such a graph. Line graphs are commonly produced from actual data as a display feature to enhance the detection of trends in the data. Mathematically speaking, the actual graph remains the plot of the ordered pairs in the function. (-5, 4) (-2, 1) (4, 1) Scatterplot … and now? Line graph Functions 7/9/2013 26 Functions 7/9/2013 26

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**Function Graphs • • • • • f = { (x , y) | y = 2x – 1 } y**

Basic Properties f = { (x , y) | y = 2x – 1 } y • (4, 7) • (3, 5) The Graph of f • (2, 3) Function Graphs For those functions that do have graphs, we can easily relate the set of ordered pairs to points in the Cartesian plane. This should be familiar territory for most students. The animation extracts the formula for functional values from the set that represents the function. For selected values of the independent variable x the corresponding functional value, that is the dependent variable y, is calculated. Both values are then projected to the point in the plane having coordinates ( x, y ) and a “point” is placed there. This is repeated for successive values of x , in this for -1, 1, 2, 3, and 4. With five points showing in the graph of f, we draw the line representing all the points in the graph and suppress the previously plotted points. The formula for the function, with no express restrictions implies the domain is the largest set of real numbers for which the formula makes sense, in this case the set of all real numbers. The line, with infinitely many points, is appropriate for representing the graph of the function. This is not considered to be a “line graph” as seen in connecting a finite set of points. • (1, 1) x • (-1, -3) Functions 7/9/2013 27 Functions 7/9/2013 27

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**Finding Domain and Range**

Basic Properties Consider: f = { (x , y) | y = 2x – 1 , 0 x 4 } y • The Graph of f (4, 7) Note: For THIS function, the domain and range are closed intervals Range = [ -1, 7 ] = { y | -1 y 7} Finding Domain and Range Here we start with the same formula as before, but alter the function f by changing its domain. Since the domain here is the interval [0, 4] the graph produced by the formula is a line segment, instead of a complete line. This is done to emphasize the role of the domain in forming a function. To extend the graphical representation, we can project all the points of the graph onto the x-axis. The projected image, the interval [0, 4] is exactly the domain of the function. Similarly, we project all the points of the graph onto the y-axis, where the projected image is the interval [-1, 7] , which is exactly the range of the function. Discussion can then follow about whether domains and ranges are always closed intervals, and if not, why not. The opportunity is also taken to show the representation of the intervals as sets of points. x • (0, -1) Domain = [ 0, 4 ] = { x | 0 x 4 } Functions 7/9/2013 28 Functions 7/9/2013

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**The Language of Functions**

Basic Properties 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 Notation Since the language of algebra, and other branches of mathematics, makes use of some commonly used symbols to describe functions, it worth the effort to explain functional notation. Again, the ordered pair concept lends itself well to the introduction of the notion of independent and dependent variables. The distinction between the name of a function and the function itself is made, since there is often confusion about which is which. The connection of the notion of functional value with the dependent variable presages the introduction of the notion of the graph of a function, which is often another point of confusion in the student’s mind. Once these concepts are laid out clearly, most students have little trouble in discussing and using the concepts correctly. Computations that later follow can then be placed in proper perspective. Functions 7/9/2013 29 Functions 7/9/2013

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**The Language of Functions**

Basic Properties 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 Notation Since the language of algebra, and other branches of mathematics, makes use of some commonly used symbols to describe functions, it worth the effort to explain functional notation. Again, the ordered pair concept lends itself well to the introduction of the notion of independent and dependent variables. The distinction between the name of a function and the function itself is made, since there is often confusion about which is which. The concept of function includes the notion of independence of domain elements, which can be chosen without regard to any other values. The dependence of each range value on the choice of domain value allows the domain value to determine the corresponding range value. That is, if you know the domain value, then you also know the corresponding range value. The converse is not true, except for one-to-one functions, which will be studied later. In general, knowing a range (output) value does not determine which domain (input) value is related to it, since there might be several output values related to the same input. The connection of the notion of functional value with the dependent variable presages the introduction of the notion of the graph of a function, which is often another point of confusion in the student’s mind. Once these concepts are laid out clearly, most students have little trouble in discussing and using the concepts correctly. The computations that later follow can then be placed in proper perspective. … OR … output is a function of input Functions 7/9/2013 30 Functions 7/9/2013

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**Functions as Tables Example**

Basic Properties 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 Functions as Tables The notion of ordered pairs as fundamental to the concept of function can be represented in forms other than an explicit set of ordered pairs. One such representation is a table which matches values from the domain of the function with elements of the range while preserving the property of uniqueness of range element for each domain element – that is, each domain element is paired with one and only one range element. In this example, knowing the tuition does not always determine the number of hours, since tuition of $2800 is the tuition several choices of hours. That is, knowing the tuition is $2800 does not tell you whether the number of hours is 12, 15, 18 or some other number above 12. Hence, H is NOT a function of T. Conversely, however, knowing the number of hours tells us precisely what the tuition will be, even though several choices of hours produce the same tuition. Hence, T is a function of H. The number of hours determines the tuition. Functions 7/9/2013 31 Functions 7/9/2013

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**Functions as Tables Example Question: Is H a function of T ?**

Basic Properties Example T H Question: Is H a function of T ? Does T determine H ? No So … H is NOT a function of T Question: Is T a function of H ? Does H determine T ? Functions as Tables The notion of ordered pairs as fundamental to the concept of function can be represented in forms other than an explicit set of ordered pairs. One such representation is a table which matches values from the domain of the function with elements of the range while preserving the property of uniqueness of range element for each domain element – that is, each domain element is paired with one and only one range element. In this example, knowing the tuition does not always determine the number of hours, since tuition of $2800 is the tuition several choices of hours. That is, knowing the tuition is $2800 does not tell you whether the number of hours is 12, 15, 18 or some other number above 12. Hence, H is NOT a function of T. Conversely, however, knowing the number of hours tells us precisely what the tuition will be, even though several choices of hours produce the same tuition. Hence, T is a function of H. The number of hours determines the tuition. This is exactly what a function does, so yes, this does fit the definition. Yes So … T is IS a function of H Question: Does this last fit the definition ? Functions 7/9/2013 32 Functions 7/9/2013

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**Graphs, Sets and Tables Ordered Pairs and Functions**

Basic Properties Ordered Pairs and Functions f = { y = 2x + 1 } (x, y) Table Input Output Input Output (x, y) x y y ● ● Graphs in General We can use graphs to determine output values that correspond to given input values. From the graph we can identify ordered pairs of input/output values as the coordinates of points on the graph. Each ordered pair belongs to a set of ordered pairs representing the relation underlying the graph. If no two of the ordered pairs have the same first component (i.e. no vertical line cuts the graph more than once), then the relation is a function. If there is a formula associated with the function, then the ordered pair is a member of the set that is the function with the symbolic representation, as illustrated. In practice the ordered pairs might be listed in a table of input/output values, as shown. Whether graph, table or formula is used to represent the function, the central concept is based on the set of ordered pairs that is the function. In practice all these types of data representation are used. In some cases, notably for non-numeric data, graphs simply don’t exist so that tabular data is all that is available. In some cases graphs are produced mechanically or electronically from data collection and there is no formula but the data can be sampled and collected in tables for convenient calculation of approximate values. If the function originates in a design effort there is probably a formula representing the function that describes a desired shape or physical action. The point here is that all of these representations of function can be useful but the they all stem from the central concept of function as a set of ordered pairs, no two of which have the same first component. (x, y) Ordered Pair ● x Functions 7/9/2013 33 Functions 7/9/2013

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**Graph and Functions Vertical Line Test**

Basic Properties Vertical Line Test No vertical line touches the graph of a function at more than one point x = x1 ● (x1, y1) ● ● ● ● Vertical Line Test We can test a graph, to see if it represents a function, by simply checking to see if any vertical line cuts the graph in more than one place (it might not cut it at all). In the first example we can easily see that every vertical line either misses the graph completely or cuts it exactly once. So this graph does represent a function. No vertical line can cut the graph twice, so the graph does indeed represent a function. For the second example, we see vertical lines that cut the graph more than once. For the line x = x1 we find points on the graph such that more than one output value (on the vertical line) is paired with the same input value on the horizontal axis. That is, there are at least two distinct ordered pairs (x1, y1) and (x1, y2) in the function that have the same first component, namely x1. In such a case, the graph does not represent a function since there are points on the graph representing ordered pairs that relate an element of the domain to two distinct elements in the range. This violates the definition of function. In the third example, the graph of a linear function, we see clearly that every vertical line cuts the graph of the linear function exactly once, verifying that the graph does represent a function. ● ● ● (x1, y2) Function ? Function? Function? YES NO YES Functions 7/9/2013 34 Functions 7/9/2013

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**Graph and Functions Vertical Line Test Examples**

Basic Properties Vertical Line Test Examples Does any vertical line cut the graph more than once ? x y x y x y y = x2 ● x = y2 ● ● ● ● Vertical Line Test: Examples In the first example each vertical line cuts the graph in one and only one point. Hence, the graph does represent a function. Note that the graph is not the function, however, just a visual representation of it. Remember: not all functions have graphs. The second example simply turns the parabola on its side (by interchanging x and y) to produce a graph that does not represent a function of x, since there are clearly lines that cut the graph in more than one point. The graph, however, does represent a function of y. By removing the lower branch of the graph in the second example, we have the graph in the third example. Since no vertical line cuts this graph more than once, the graph is that of a function. ● x = y2, y > 0 Function? Function? Function? YES NO YES Functions 7/9/2013 35 Functions 7/9/2013

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**Relations a b m c h j k Domain Range**

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

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**Relation As A Set { } Relation: Any set of ordered pairs Notaton:**

Basic Properties Relation: Any set of ordered pairs Notaton: { } ( a , b ) , ( c , d ) , ( e , f ) , ( g , h ) Domain: { a , c , e , g } Set of all first components Relation as a Set of Ordered Pairs The introduction of relation as a set of ordered pairs provides a lead-in to the definition of function. This approach to the concept of function has the advantage of providing a concrete mathematical object for the student to focus on as a function. It tells the student what kind of thing a function is. Focusing on the concept of domain and range embodies the notion of relating one set of objects to another in a unique way and provides a simple way to view both the domain and range of a function. While allowing for unique combinations of elements mapping to a range element, the notion of relation allows for restricting such combinations to a single element – the concept of function. Range: { b , d , f , h } Set of all second components NOTE: DOMAIN and RANGE are SETS of objects Functions 7/9/2013 37 Functions 7/9/2013

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Examples Basic Properties Relations A = { ( a , b ) , ( c , d ) , ( c , f ) , ( g , b ) } Domain of A = Range of A = { a, c, g } { b, d, f } Relations These examples should help clarify the connection between domain values and range values for relations. The ordered pair is the key concept here, since each ordered pair shows which domain element is related to which range element. Note that domain element c is mapped to two distinct range elements, d and f. That is, at least two distinct ordered pairs have the same first component, so this relation is not a function. Functions 7/9/2013 38 Functions 7/9/2013

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Examples Basic Properties Relations B = { ( -6 , 7 ) , ( 3 , 0 ) , ( 2 , 7 ) , ( -2 , 0 ) } Domain of B = Range of B = { -6, -2, 2, 3 } { 0, 7 } Relations These examples should help clarify the connection between domain values and range values for relations. The ordered pair is the key concept here, since each ordered pair shows which domain element is related to which range element. In this case, each domain element is mapped to one and only one range element, so this relation is also function. Note that ordered pairs (3, 0) and (-2, 0) have the same second component, as do pairs (-6, 7) and (2, 7). This is allowed in a function. Functions 7/9/2013 39 Functions 7/9/2013

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Examples Basic Properties Relations S = { (1493, Bob), (3872, Sally), (2840, Bob), (5492, Mary) } Domain of S = Range of S = { 1493, 3872, 2840, 5492 } { Bob, Sally, Mary } Relations This example shows a mapping of numbers, possibly id-numbers or account numbers, to names. Because elements such as id-numbers and account numbers are generally unique to the individual, each number maps to one and only one range element, i.e. to one and only one name. Conversely, however, each range element may have more than one domain element mapped to it. For example, both 1493 and 2840 map to Bob. Thus, there are two ordered pairs with the same second component. All such combinations are allowed in a relation, and this last is also allowed in a function. Note that no two ordered pairs have the same first component, a requirement for a function. Functions 7/9/2013 40 Functions 7/9/2013

41
**The Word “RANGE” The two meanings of RANGE**

Basic Properties 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 Meanings of Range As in the English language some words have more than one meaning. The distinction is normally drawn from the surrounding context. In this case, the word range is used as a number representing a statistic associated with a set of data, namely the difference between largest and least values. However, the word range is also used as a target set for a relation, providing elements to include as second components in the ordered pairs of the relation. So, in one case the word range refers to a number and in the second to a set of objects. The two notions are not related. Functions 7/9/2013 41 Functions 7/9/2013

42
**Graphical Presentations**

Basic Properties Graphical Presentations One-variable data in one dimension Example: -21, -13, -8, 3, 5, 9, 12, 22 10 20 -20 -10 Graphs: Geometric Presentations Graphs represent a way to visualize data in a geometric or graphical way to provide a sort of “picture” of the data. For one-variable numerical data we simply plot the values on the real number line as a dot. It is important to note that the things we are plotting here are numbers. The graph of the data is just the set of points in the plane represented by the numbers in the table. Note that the graph is a geometric object – the set of points on the line. In this case there is no “curve” or “line” that is the graph, each of which would require plotting infinitely many points. Note that graphs of this type are always read from left to right, that is, in order of increasing independent (or “input”) variable values. A plot of numbers Functions 7/9/2013 42 Functions 7/9/2013

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**Graphical Presentations**

Basic Properties Graphical Presentations Two-variable data in two dimensions Example: A plot of ordered pairs x y x y 2 -4 5 -2 8 6 (8,6) (-1,4) Graphs: Geometric Presentations Graphs represent a way to visualize data in a geometric or graphical way to provide a sort of “picture” of the data. For two-variable numerical data we plot the ordered pairs from the relation represented by the data. The plot is normally made on a copy of the Cartesian (rectangular) coordinate system, where one of the variables (the independent variable or first component) is plotted against the horizontal axis and the other variable (the dependent or second component) is plotted against the vertical axis. The coordinates of each point are just the ordered pairs from the table of data. The graph of the data is just the set of points in the plane represented by the ordered pairs in the table. Note that the graph is a geometric object – the set of points in the plane. In this case there is no “curve” or “line” that is the graph, each of which would require plotting infinitely many points. We can, however, “connect the dots” to form a “line graph” which gives a sometimes useful view of how the points are related. Note that graphs of this type are always read from left to right, that is, in order of increasing independent (or “input”) variable values. (-7,2) (5,-2) (-5,-3) (2,-4) Functions 7/9/2013 43 Functions 7/9/2013

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**Think about it ! Basic Properties Functions 44 7/9/2013 Functions**

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