Fundamentals of functions
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1. CHAPTER 4 FUNDAMENTALS OF FUNCTIONS 4 3 2 Transforming the parents As we have seen many times already a graph can be particularly helpful way to see and therefore understand a mathematical relationship One of the skills we ll develop as we go forward in this course is the ability to connect features of the equation of a certain function to features of its graph Over time and with practice we ll get better at picturing the graph of a function in our mind s eye without having to draw the graph on paper Drawing graphs will still be informative of course but some features of the graph will start to jump out at us simply from looking at the equation of the function Not far from the tree startup exploration Study the three sets of animations depicted in figs 4 3 to 4 5 Each set of images shows a way in which we can change the parent equation and produce a change in its graph Figure 4 3 shows what happens when we multiply the parent function by a constant value Figure 4 4 shows what happens when we add a constant value to the parent Figure 4 5 shows what happens when we add a constant value to the agrument the x value before that value is used as input to the parent function Study the animations or experiment with the interactive versions at algebranomicon org Then write a sentence or two summarizing what is happening in each case a The images will only appear animated when this PDF document is vi2. 3 4 We can also define a specific domain for convenience On an assignment for example we might be asked to evaluate the function y 4x 5 for the x values 1 0 and 1 We did something like this in example 3 1 This is another case of limiting the domain 4 2 2 Regarding output values In addition to knowing about input values it is often important for us to know what sorts of numbers we can expect as output from a certain function We call the set of all output values the range of the function The set of all output values from a function This is the set of all y values or all possible values of the dependent variable The function f x x can accept any number as input but only non negative numbers ever come out We say that the range of f is non negative real numbers In the case of g x L notice that the only way for a fraction to equal 0 is if the numerator is equal to zero No matter what nonzero input value we put into the function g we will never get 0 as the output The range of g is nonzero real numbers 6 Did you know that the so called Hawaiian pizza a pizza topped with pieces of ham and pineapple is not actually a Hawaiian invention Several news organizations agree that this pizza originated in Ontario Canada 86 CHAPTER 4 FUNDAMENTALS OF FUNCTIONS Piecewise definition An interesting way to control the output of a function is to give it different definitions depending
3. 87 CHAPTER 4 FUNDAMENTALS OF FUNCTIONS After all since the integers include all of the natural numbers and all of the opposites of the natural numbers it must be that the set of integers has twice as many members right Nope The set of natural numbers and the set of integers are exactly the same size Think of it this way the function z defined above pairs up natural numbers and integers If give you a natural number input can you tell me what integer output it partners up with If give you an integer output can you tell me what natural number input it partners up with If so then the two sets match up exactly and neither set has any leftovers They must therefore be the same size Mind blown 88 CHAPTER 4 FUNDAMENTALS OF FUNCTIONS 4 3 Three families of Algebra 1 The majority of Algebra 1 will focus on the in depth study of three key families of functions the linear family the exponential family and the quadratic family We ve encountered each of these families before and now we will recap all that we ve learned about them Though this is our first formal introduction you might be surprised to discover how much you already know about the families 4 3 1 Parent functions For a given function to be considered a member of a certain family it must share the characteristics of other members of that family as seen in its graph its equation and the patterns it exhibits in a table of values These f
4. mothers in a legal or emotional sense but a person always has exactly one biological mother 78 CHAPTER 4 FUNDAMENTALS OF FUNCTIONS 600 500 400 300 Depth ft 200 100 0 15 30 45 60 75 90 Elapsed Time min Figure 4 1 A graph of depth over time Impossible Function A function is a special type of relation in which the ordered pairs have the following property each x value is paired with one and only one y value The one and only one requirement is what makes a function special The graph in fig 4 1 violates this requirement certain x values have more than one associated y value look at x values between 15 and 45 minutes So this graph does not depict a function The sequences we studied in chapter 2 are functions it would be silly to suggest that a sequence could have more than one value as its third term Depending on how we think about number machines we might expect that a machine would act predictably and always produce the same output for a given input A number machine that behaves in this way defines a function Representing relations Ther are several ways to represent a mathematical relation We might represent a relation using e Agraph This representation gives us a visual of the relationship between input values and output values e A table of values This representation gives us an organized list of input values and their corresponding output values Related fo
5. output Or we might define the name the pet relation which accepts a human as input and outputs other animals Note that there is a difference between these two relations we have just defined Not all people own pets so some inputs to the name the pet macine have no output Plus some people own multiple pets so some inputs to the name the pet machine have multiple outputs On the other hand the name the mother machine iS guaranteed to always produce a unique output every person has a biological mother and it is impossible for a person to have more than one biological mother We saw something similar in section 3 5 where certain graphs could be drawn in a way that would suggest an impossible situation For example we drew a graph shown in fig 4 1 that suggested Yeardleigh s submersible could at three different depths simultaneously This is as ridiculous as the idea that a person could have more than one biological mother Relationships which follow this very natural rule any given input produces a unique output have a special status in algebra They re called functions There are other rules that we could write based on the two data points we were given and perhaps you thought of different rules than the ones we describe here Can you find alternative explanations for how each machine works Be creative We don t deny that a family with two mothers is still a family It is true that a person can have multiple
6. Linear exponential and quadratic functions all have all real numbers as their domain We can always use any number as input This is convenient The range of a linear function is also pretty simple it is all real numbers As we saw as we transformed linear functions the a non horizontal line will eventually stretch up or down as high or as low as you want to go on the y axis The ranges for exponential and quadratic functions are a little more complicated but we can see what is happening by examining their graphs Notice for example that the graphs of the parents y 2 and y x never dip below the x axis This feature of the exponential and quadratic families will always be retained in some form when we transform these equations the range will always be limited by some minimum or maximum value The quadratic parent passes through the origin and so the range of y x is all real numbers greater than or equal to zero We might write that y is any real number such that y gt 0 In fact every parabola has a point called its vertex that is either its lowest point if the parabola opens upwards or its highest point if the parabola opens downwards We will learn more about identifying and describing the vertex of a parabola when we study the quadratic family in in depth When it comes to the exponential parent we mentioned that is has an asymptote at y 0 This means that the function approaches but never reaches
7. On two occasions have been asked Pray Mr Babbage if you put into the machine wrong figures will the right answers come out am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question Charles Babbage English inventor of the first mechanical computer Chapter 4 Fundamentals of functions We have seen a number of relationships so far the relationship between distance and time of a moving object for example or the relationship between a number s position in a sequence and that number s value A function is a certain kind of relationship that is of key importance to us in algebra and indeed throughout mathematics In this chapter we ll get an overview of functions and their different representations Then in the rest of the course we will look closely at three specific kinds of functions 4 1 Mathematical relationships Number machines startup exploration A number machine accepts numbers as input and produces numbers as output When machine A receives the number 8 as input it produces the output 28 When this machine receives 12 as input the output is 42 Machine B has a different mechanism for producing output values When machine B receives the number 8 as input it outputs 15 When it receives 12 as input the output is 23 What do you suppose each machine will output when given the number 7 as input Write a sentence explaining how you believe each mac
8. at each step So this is the feature we see in the table of a linear function when the x values increase by a fixed amount the y values increase by a possibly different fixed amount There are two key transformations of the parent function Multiplying by a constant value as in y a x changes the steepness of the graph Certain a values cause the graph to flip over the x axis and show a trend that goes in the opposite direction we get a graph that has been reflected over the x axis Adding a value as in y Xx c shifts the graph upwards or downwards Viewed from a different perspective we could also say that this shifts the graph left or right Up down shifts and left right shifts look the same for a straight line The distinction between these two transformations is more clear in the other families 4 3 4 Fundamentals of exponential functions The second family of functions we will explore in this course is the exponential family In fact we will begin our study of this family in Algebra 1 and then return to it again in Algebra 2 An exponential equation is one in which the variable x appears as the exponent hence the name The graph of an exponential function is a smooth curving shape that we sometimes describe as J like This is only an approximation though since an exponential graph may sometimes look more like a backwards J or an upside down J Exponentials have the interesting property that they get almost
9. eatures are shown in their most basic form by the parent function of each family The linear parent is y x and the quadratic parent is y x In this course we will consider y 2 to be the parent function for the exponential family though other choices are possible Linear Family Exponential Family Quadratic Family f x x f x 2 f x x Figure 4 2 The parent functions of the three families Compare the equations for each of the parent functions How are they the same How are they different Compare the graphs How are they the same How are they different When presented with a new equation of graph what features might we look for to decide which family it belongs to In section 4 1 we learned techniques for identifying whether a given relation is a function One of our tasks going forward will be to learn techniques for distinguishing the function families from one another 8 A more natural choice for the exponential parent is the function y e where e is Euler s number yes that s the same Euler as the function notation guy Euler s number has many important connections to the family of exponential functions but those will have to wait until later By the way e like m is an irrational number e amp 2 71828 18284 59045 23536 02875 These are not the only three families of functions out there of course Other families of functions will become the subject of study in Algebra 2 and beyond 89
10. em in diagram 2 is not that multiple arrows point to each of the output numbers It s OK for multiple arrows to go toward a certain output value The problem is that the input value 2 has multiple arrows leaving and going to different targets both 7 and 9 82 CHAPTER 4 FUNDAMENTALS OF FUNCTIONS 4 1 3 Equations of functions The number machine metaphor is often helpful when thinking about a function A number an x value goes into the machine the machine s rule works on the number and another number comes out of the machine the y value Most of the time when we have a particular rule and a particular input we get a single predictable output For example the equation y 3x defines a function We can substitute in 5 for x The function machine multiplies 5 by 3 and outputs 15 This generates the ordered pair 5 15 and we say that 5 maps to 15 If we choose other values of x we can generate more ordered pairs Not every equation defines a function though In order to be a function every input value must produce one and only one output value In Algebra 2 we will study many interesting and important mathematical relationships that are not functions We will learn for instance how to write the equation for a circle and a circle is not a function Can you explain why not Our focus in Algebra 1 though will be on three main families of functions and our rules will all be of the form y equals
11. er the function machines below fe lxl ax What sorts of values are allowed as input to each of these machines What sorts of values will each one produce as output A blender is a machine that has limited operating parameters Anything we put into the blender comes out blended in some way But of course there are some things we can t put in a blender a piano for example Similarly we wouldn t put bread into a blender hit the button and expect toast to pop out That kind of output is associated with a different machine 4 2 1 Regarding input values An important component of the definition of any function is a specification about what sorts of numbers it can accept as input In the language of mathematics we call the set of all legal input values the domain of the function Domain The set of all allowed input values to a function This is the set of all allowed x values or all possible values of the independent variable We have two functions in the startup exploration The first function f x x is very welcoming any real number can be used as input to this number machine whole numbers fractions positive numbers negative numbers it doesn t matter The other function is a little more finicky 5 We ll pause here to note that blended is defined rather loosely smoothies come out blended ice comes out crushed chickpeas come out pur ed magazines come out
12. ewed using Adobe Reader 4 3 3 Fundamentals of linear functions The first family we will study in depth is the family of linear functions Here we summarize the key features of this family all of which will return in future chapters We will learn several different forms for the equation of a linear function The feature they all share is that the variable x has no exponent or rather the highest power of the variable x is 1 For example the parent of this family y x is the same as y xt That s a phantom 1 there in the exponent and we don t usually write it The graph of a linear function is a straight non vertical line Can you explain why we have to exclude vertical lines from this family of functions We sometimes also exclude horizontal lines from this family a function of the form y a number is a horizontal line for example y 6 1 10 In fact horizontal lines can be considered their own family the family of constant functions 90 CHAPTER 4 FUNDAMENTALS OF FUNCTIONS Figure 4 3 Comparing the parent function f x with offspring a f x Figure 4 4 Comparing the parent function f x with offspring f x c Figure 4 5 Comparing the parent function f x with offspring f x b 91 CHAPTER 4 FUNDAMENTALS OF FUNCTIONS Recall that arithmetic sequences are in the family of linear functions Arithmetic sequences have a rule that involves repeatedly adding on a constant difference
13. hine works The number machines described in the startup exploration relate certain input values to certain output values Mathematically speaking a number machine defines a relation T7 CHAPTER 4 FUNDAMENTALS OF FUNCTIONS Relation A relation defines how certain numbers or other objects are connected to other numbers or objects We can think of a relation as a collection of ordered pairs x y meaning x is related to y We might think of a number machine as a set of pairs of numbers input output Machine A in the startup exploration is defined by the pairs 8 28 and 12 42 We sometimes say that machine A maps 8 to 28 and that 12 is mapped to 42 One possible explanation is that the machine multiplies the input value by 3 5 to produce the output value In this case we d expect the input 7 to produce the output 24 5 In other words 7 24 5 is a third point associated with machine A For machine B we are given the points 8 15 and 12 23 One somewhat complicated explanation is that the machine doubles the input takes the absolute value of the result and then subtracts 1 In this case we d have 7 13 as another input output pair for machine B We can define relations that compare things other than numbers For example we might define the name the mother relation that accepts a person as input and which gives that person s biological mother as
14. l credit though A graph fails the vertical line test if any vertical line even just one crosses the graph in more than one point This happens also with the graph in fig 4 1 That graph fails the vertical line test for some vertical lines Therefore the relationship depicted is not a function 80 CHAPTER 4 FUNDAMENTALS OF FUNCTIONS Example 4 2 Which of the scatter plots below if any represents a function How do you know You may have to look closely Plot A Plot B Plot C Solution Plots A and C both pass the vertical line test and therefore represent functions Plot B fails the vertical line test because it fails for the vertical line at x 3 Note that we don t care if different x values map to the same y For example in plot C all of the x values map to the same y value That s OK In other words a function can fail the horizontal line test without penalty For example the name the mother function does not pass the horizontal line test because two different inputs people can have the same output mother In fact it is not at all uncommon for two different people to have the same mother they re called siblings 4 1 2 Functions from points A scatter plot gives a visual representation of a collection of ordered pairs If the ordered pairs are presented in an alternative format say a table of values we can use the same reasoning to determine whether the given relation is a func
15. nent on the variable x is 2 The equation may also contain an x term but that s the same as xt so the highest power of x is still 2 The graph of a quadratic function is a smooth U like graph called a parabola Of course under certain circumstances the U opens downwards The transformations of the parent function are perhaps more clear with this family Multiplying by a constant y a x changes the steepness of the graph And in the case that a is negative controls whether the graph opens upwards or downwards Adding to the parent function y x c produces a clear vertical shift and adding to the argument y x b produces a clear horizontal shift Interestingly the horizontal shifts appear to be backwards compared to what we might expect Adding to the argument shifts the graph to the left whereas subtracting from the argument or adding a negative value shifts the graph to the right Can you spot this unsual behavior in the linear and exponential families too Recall from our study of sequences that quadratic patterns show a constant second difference In other words they have a recursive rule that states Start with A add to each previous term the members of some arithmetic sequence This constant second difference will be the feature that we look for in a data table 4 3 6 Domain and range of the three families It s quite easy to identify the domains of the three main families of Algebra 1
16. on the input For example suppose we define a function like so x if x is positive f x 4 x if x is negative O if x is equal to 0 a given input value x If the input x is positive then the output is simply x itself If the input is negative then the output is the opposite of x so again we get a positive value Finally in the third case if the input is 0 then we get 0 as the output Do you recognize this function This case by case definition is really just a complicated way of saying f x x Here s a more interesting example Suppose we define a function whose domain is the non negative integers as follows n fni if n is even 2 z n n 1 2 This function produces different output depending on whether the input value is even or odd Let s try the first if n is odd few input values to see what happens What happens when we input 0 Zero is an even number so the first case applies and r z 0 0 0 5 One is an odd number so the second case applies 1 1 2 E 27 Two is even and three is odd So we have 2 z 2 a 1 and z 3 This function produces the sequence 0 1 1 2 2 3 3 4 4e which is a list of all the integers In other words this machine turns the natural numbers N plus zero into the set of integers Z This function suggests a shocking idea Most people would say that there are more integers than there are natural numbers
17. rms include a mapping diagram or just a collection of ordered pairs e An equation This representation gives us a rule for turning input values into output values the way a number machine might do 79 CHAPTER 4 FUNDAMENTALS OF FUNCTIONS Since functions hold a special position in the universe of all relations our first task is to determine whether a given mathematical relationship is in fact a function To do this we have to check it against the definition of what it means to be a function In other words we must make sure each x value corresponds to one and only one y value 4 1 1 Graphs of functions In order for a graph to be a function it must avoid the situation in which multiple y values stack up above a particular x value A handy way of remembering this requirement is that the graph of a function passes the vertical line test The vertical line test states that for the graph of a function any vertical line drawn on the graph will intersect the graph exactly once Example 4 1 Which of the graphs below if any represents a function How do you know Graph A Graph B Graph C BEP Solution Only graph A represents a function We can draw vertical lines on graphs B and C which intersect the graph at more than one point This means that those x values have more than one y value violating the definition of function Note that in graph C certain sections of the graph pass the vertical line test There is no partia
18. shredded We consider all of these to be a kind of blended Speaking of magazines we ll also mention that there are things which one can put into a blender but shouldn t Magazines are one example Another example your hand 85 CHAPTER 4 FUNDAMENTALS OF FUNCTIONS In chapter 1 we discussed the illegality of division by zero So if we try to put 0 into the function g x L then we re going to have a problem g 0 that s the function g evaluated at x 0 is undefined So our user manual for the function g has to include a note that only numbers other than 0 are allowed as input We say that the function g x has a limited or restricted domain There are other circumstances where we will want or need to limit the domain For example a domain can be limited by the context of a problem we are solving If we write an equation that shows distance as a function of time it usually makes sense to limit the domain to only include positive time values If we have a function which maps number of pizza toppings to price of the pizza then there is probably an upper limit on the domain there are only so many pizza toppings available A sequence is a function whose domain is limited to the set of natural numbers the positive integers We saw this in chapters 2 and 3 a sequence can have a first term and a second term but no one and a halfth term We exclude from the domain of a sequence all numbers except 1 2
19. some expression in terms of x 4 1 4 Function notation We can thank Swiss mathematician Leonhard Euler for the concept of a function He was the first to coin the term function and was the first to use a handy way of writing a function called function notation Up until now we have used y something in terms of x to define all of our rules Euler s function notation looks a bit different but it is quite helpful in certain situations The generic form of function notation is f x something in terms of x In this case we simply replace y with the notation f x which is read aloud as f of x or f as a function of x In its generic form the letter f is used to name the function We use the letter f of course because it stands for function And as usual we use x to represent the independent variable What s nice about this notation is that it allows us great flexibility to use other letters or symbols that can be more descriptive For example if we wanted to describe how the height of a bouncing ball changes over time we might want to let the variable t represent time and then name the function h for height So we could write a function h t some expression in terms of t That s read h of t which does a pretty good job of capturing the idea that we re describing height in terms of time We might write a function like d t 60t which described distance traveled d as a func
20. that value So the range of y 2 is all real numbers strictly greater than zero We might write that y is any real number such that y gt 0 We will learn more about what s going on with this asymptote when we study the exponential family 93 CHAPTER 4 FUNDAMENTALS OF FUNCTIONS Chapter summary In this chapter we have brought together a number of different concepts Our work with sequences graphs functions and transformations has been organized under three key headings the three families that we will study closely in Algebra 1 With the next chapter we begin to focus on the linear family the most logical place to start In one sense the linear family is the easiest family to understand and so it makes for a good training level At the same time the linear function is an important structural concept in mathematics that gives rise to a whole branch of study called appropriately enough inear algebra In any case studying the linear family will give us the chance to learn a whole collection of new algebraic tools and techniques that we will use throughout the course Onward 94
21. tion 3 The horizontal line test is a thing and passing or failing the horizontal line test does have a mathematical meaning A discussion of this will have to wait until later however 81 CHAPTER 4 FUNDAMENTALS OF FUNCTIONS Example 4 3 Which of the data sets below if any represents a function How do you know Set A Set B Set C x y x y 4 1 2 6 1 2 0 4 1 6 i 4 Li 7 2 3 0 4 0 2 4 3 1 5 2 5 i 0 3 4 2 1 0 3 0 0 6 Solution Set A is a function There are no repeated x values Set B is not a function The x value 1 appears twice and with two different related y values 0 and 2 This violates the definition of function Set C is a function The x value 0 appears twice but it is mapped to the same value in both cases A mapping diagram is a similar way of representing a relationship between specific input and output values Here we draw two regions represented by a box or an oval In one region we list the input values in the other region we list the output values Then we draw arrows to show which input maps to which output Example 4 4 Which of the mapping diagrams below if any represents a function How do you know Diagram A Diagram B i Solution The first mapping is a function Each x value maps to exactly one y value The second mapping is not a function the value 2 maps to both 7 and 9 Note again that the probl
22. tion of time t This is exactly the same equation as y 60x but we ve changed the names of things to better match the context 4 The surname Euler is German and so it is pronounced OIL er and not YOO ler 83 CHAPTER 4 FUNDAMENTALS OF FUNCTIONS Evaluation using function notation Function notation has another convenience We can use the notation to indicate a specific choice of value for the independent variable Earlier we said Evaluate the function y x 4 when x 3 This was fine but function notation gives us a way to shorten these instructions For example if we have f x x 4 then we might want to evaluate f 3 which is said aloud f of three The x in the function notation was replaced by 3 which is exactly what is means to evaluate the function at that value The notation is telling us to input 3 for x and find the output value So if f x x 4 then f 3 3 4 9 4 5 We will use function notation and y notation through this course The flexibility of function notation means that it is the preferred way of writing functions in higher level mathematics courses 84 CHAPTER 4 FUNDAMENTALS OF FUNCTIONS 4 2 Domain and range Machines meaning real mechanical devices are physical things that have physical limitations Number ma chines meaning functions often have similar operational restrictions The ins and outs of functions startup exploration Consid
23. vertical on one side and almost horizontal on the other side For example one side of the parent function y 2 gets infinitely large while on the other side we have tinier and tinier fractions that get closer and closed to zero but never actually disappear Mathematically speaking we say that the function approaches an asymptote at the flat horizontal portion The parent function has a horizontal asymptote at y 0 other functions in this family may have their asymptote at a different position Recall that geometric sequences are in the family of exponential functions Geometric sequences have a rule that involves repeatedly multiplying by a constant ratio This is the feature we look for in a table Multiplying the parent by a value as in y a 2 changes the steepness of the graph and if a is negative causes it to be reflected over the x axis Adding a value to the parent function y 2 c causes a shift up or down Can you predict where the function s asymptote will be in this case If we add to the argument y 2 we get a shift left or right 4 3 5 Fundamentals of quadratic functions We will study the quadratic family last in this course In fact it will be our gateway into the study of techniques that will become helpful as we learn about other families and types of functions in Algebra 2 92 CHAPTER 4 FUNDAMENTALS OF FUNCTIONS The key feature of a quadratic equation is that the highest expo
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