Functions - Definitions and first examples and properties

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1 Functions - Definitions and first examples and properties Jean-Luc Bouchot jean-luc.bouchot@drexel.edu February 4, Definition and examples 1.1 Preliminaries What is a function? Besides its mathematical sense and magic we give here some details. Think of a function as a transformation, a machine. You feed your machine with something, it transforms it into something different generated only by the input. As an example, think of a MATH-101 specialized oracle. You could ask the oracle to tell you what will be the test for the final or the different quizzes. You would feed the oracle with a date, and it would return the exam. However such oracle is still rare to find To keep track of more probable situations, think of a thermometer. At a given time t (the input) it gives you the temperature T (the output). Another example comes in terms of stock market. Consider the function that returns the value (in dollars) of an action of a companie at a certain time (and date). This is yet another example of input-output relationship. Hence mathematician defined these concepts with their own words and we shall study them right now. 1.2 Definition Definition 1 (Function). A function is a rule that affect a single output (or y-values) to all of its intputs (or x-values) In other words, the output is uniquely defined by an output. Consider the following table: Time (hh:mm) 08:00 08:30 09:00 09:30 10:00 11:00 11:30 Temperature ( F) It gathers some value for the temperature at a given location and at certain times of the day. We see that to each time is associated a unique value of the temperature (at that location); this can therefore be seen as a mathematical function. Remark When asked if an object is a function, an easy way to check it is to order all the inputs from the smallest to the greatest: if one of them is doubled, and the associated y-values are different, this object can not be a function (for the output is not uniquely defined) 2. As we can see from the previous example, a function can have many times the same output. This is not a problem at all as the only condition in the definition of a function concerns the input: to one input, I can associate a single output. Exercise 1. Are the following examples possible functions? F 1 = {(1, 1.1), (2, 2.2), (1, 3.3), (0, 4.4), (3, 5.5)} F 2 = {(0, 0), (0.5, 10), ( 1, 10), ( 1.5, 20), (10, 20)} 1

2 1.3 Notation As functions often appear (and not only in mathematics!) people use rather uniform notations for them. As we have already seen, we can have functions expressed in an exhaustive way where we give the associations between inputs and outputs. A second option is to give an explicite rule that allows to compute or derive the output solely based on the input (this is for instance not the case for the temperature example). In that case, we often write y = f(x) (1) In that case, y is called the output and x the input. This x is also called a variable as it can change its value; depending for instance on the time of the measurement of the temperature. Example 1. Assure you are given the following equation: y = f(x) = x We want to know the output of the function f for the following set of input: { 1, 0, 1, 2, t}, where t is another variables, coming from, for instance, another calculus. f( 1) = ( 1) = = 2 f(0) = (0) = = 1 f(1) = (1) = = 2 f(2) = (2) = = 5 f(t) = (t) = t Exercise 2. On Earth, the force your body applies to the surface of the planet is related to your weight by the following formula: F (w) = 9.81 w (2) where w is your weight in kilograms and F is the force you exerce on Earth (in Newton). What is the value, in Newton, of the force your body creates on Earth? (Make sure to have the correct unit!) 1.4 Some examples, counter-examples and applications Give here some easy example in the form f(x) = something and some computations with that. Exercise 3. Consider the functions f(x) = 3 x + 2 and g(x) = 1.5 x + 1 and compute the following values for any non-zero h: f(x + h) f(x) f h (x) = h g(x + h) g(x) g h (x) = h Example 2. The relation y 2 = x + 1 does not define a function. Indeed the input x = 0, for instance, does not have a single output: y = 1 or y = 1. 2 Representation We have already seen how we can guess the value of the output based on an input we know the rule we use; i.e. when we have an equation for the function. However, we have also different ways to represent a function. 2

3 2.1 Tables and diagram When using tables, the data (or x and y values) are organised such that each input of the function is facing its unique output. If we cannot choose which output it should face, we are not dealing with a function. The diagram representation is somewhat similar but in this case, we do not enforce our input to face their respective outputs, but rather to really mark their relationships with an arrow starting from the input and pointing towards its output. We have already seen how to represent a function through a table in the case of the temperature, we can switch to the equivalent representation using a diagram (see the next figure) Inputs Outputs 10 : : : : : : : Note that the only condition is that every input has no more than one output. The fact that one output is linked to an input is not considered in the definition of a function. Moreover, as we can see, the output 35 has to inputs. This is also not a problem at all. 2.2 Set of ordered pairs Another pratical way to represent a function is by means of ordered pairs. Before digging into more details, let us recall (or give) some definitions Definition 2 (Pair and ordered pair). A list of two elements is called a pair and is written with the following curly brackets: {element 1, element 2}. The position of each element is not important when considering a pair (i.e. {element 1, element 2} = {element 2, element 1}) An ordered pair is a pair where the position of each element has an importance and is written as (element 1, element 2). In that case, and unless the two elements are equal, we have (element 1, element 2) (element 2, element 1). Example 3. Consider two card games. In the first one, a player pick two cards at random and sum their values (assuming, for instance that all heads have value 10). The aim of the game is to have the highest score, or at least higher than the one of your opponent. In that case, the order in which you pick the cards is completely irrelevant therefore we would consider simple pairs of cards. In another game, you pick first a card and should then guess whether the next one you will pick as a value greater, smaller or equal to the first one. In that case, the order in which you pick the cards is relevant. We would represent your hand as an ordered pair. Definition 3 (Sets). A set is a container in which you may pot any objects of the same nature. They are also represented with curly brackets { some stuff } 3

4 Remark 2. While the definition seems somewhat hard to read you have to understand the importance of making it so general. Indeed, we have already seen that the notation { } is used to represent pairs. So an alternative definition for a pair could be a set containing two elements. In the previous example, the elements contained in that set were cards. We will see in the next definition that a function can be represented by a set of ordered pairs (i.e. a set that contains ordered pairs). A last remark is to note that a set can also be empty (which will rarely be the case in our course) in which case we write Now we can go back to our table or diagram representation of a function and write it a set of ordered pairs in which the first element corresponds to the input and the second one to its associated output. 2.3 Graph Finally, and as maybe the most usual way to represent a function, we often make use of graphs and plots. In that case, all the pairs (x, y) of input x and their output y are displayed on a single plane. This plane is called either called the x y plane (due to the fact that it directly associates the output y of a given input x) or the Cartesian plane (thanks to René Descartes). The plane is separated in 4 quadrants usually numerated from I to IV starting on the upper right corner and following a counter clockwise direction (See figure 1 for an illustration). y II I Origin: (0, 0) x III IV Figure 1: The Cartesian plane and its four quadrants The idea behind the graph representation of a function is to plot all the (x, y) pairs we can. The x-values will represent the horizontal offset from the origin while the y-values correspond to vertical offset (or height). The pair (x, y) is called the coordinates of the point. If we go back to the temperature example, we can plot the different points we had (we will convert the Fahrenheit into Celsius and consider the origin at 8am to make it easier to plot): Note that you should not join the points in that case. Indeed there is no way you can infer the behavior of the temperature as a function of the time just from this scatter plot. In mathematical words, we say that the function is only defined on a certain set of values (which is called the domain, a notion we will introduce in the next section). Property 1 (Vertical Line Test). A graph is a valid representation of a function if it fulfills the Vertical Line Test ( VLT); i.e. whenever a vertical line is drawn on the graph, that line crosses the curve at most once. 4

5 T (3.5, 3.5) (3, 3.2) (2, 3) (1,(1.5, 2) 2) (0.5, 1) (0, 0) t In some cases the function is completely defined everywhere. In this case its graph is not longer drawn as a set of points scattered around the Cartesian plane but rather some kind of continuous line (note that while this will be true for most of the cases, we will see some examples of functions that have jumps or bumps in their representations) y y = f(x) x Figure 2: Example of a graph of a function defined on a continuous set of values. The points on that line are related by the fact that y (the height) = f(x) = x 3 + 1, with x the free variable, or horizontal offset 2.4 Examples and exercises 3 Domain and range Two important concepts when studying functions is there domain and ranges. In a certain way, we can understand the domain as the objects which can be fed to out function. Under the condition that the machine is being fed properly (i.e. the input stays within the domain) we are sure that the output will land in what is called the range 5

6 3.1 Definitions Definition 4 (Domain). The domain of a function f is the set of numbers that has an output when passed as an input to the function f. In other words it is the set of all possible inputs of the function. In general, the domain is denoted by the letter D (or D f in case we have different domain D f, D g, for different functions f, g, ) Definition 5 (Range). The range of a function f is the set of numbers that can be reached by at least one input passed into f. In other words, a number is in the range of f, if f maps at least one input from its domain D to this number. Property 2. Finding the domain of a function f is either 1. Listing all inputs, when the function is given as a diagram, table or set of ordered pairs or 2. Finding the largest ensemble of numbers such that all computations can be done without trouble, when f is given as an equation. Property 3. So far, only two computations cause trouble: Rule 1 Do not divide by 0 Rule 2 Do not take the square root of a strictly negative number. (this actually holds for any even root) Remark 3. Along this class, we will add other rules as we introduce new functions. More later on that topic! 3.2 Computations and examples Finding the domain of some function is not always an easy task and one needs to practice to understand and acquire all the tricks. Here are some examples one should study in details. Example 4. Let f be the function defined by the following formula: f(x) = 3x + 2. Finding the domain of f is equivalent to considering the whole line of numbers (which we write (, + )) and taking out all the problematic numbers. Here, we see that the function has neither a square root nor a fraction. Therefore, there is no problem and the domain is the whole line of numbers: D = (, + ). Example 5. Let f be the function defined by the following formula: f(x) = 1 3x+2. Once again with start with the complete line and try to find where troubles can be encoutered. In this case, we do have a fraction and applying rule 1 tells us that the denominator cannot be 0. Therefore we look for the inputs x such that the denominator 3x + 2 = 0. Once we have found it/them, we will take them out of the original domain (the complete line). 3x + 2 = 0 3x = 2 x = 2 3 Hence, the set of inputs cannot contain the value 2/3. Therefore, the domain can be described as D = (, 2/3) (2/3, 6 + ) 6

7 Example 6 (Domain and range of a function as a set of ordered pairs). This example deals with the case that a function is given as a set of ordered. Consider the following function: {(0, 1), (1, 0), (2, 1), (3, 0), (4, 1), (5, 0), (6, 1), (7, 0)} First try to convince yourself that this indeed is a function. The range can be recovered by taking the set of all first elements of all pairs: D = {0, 1, 2, 3, 4, 5, 6, 7} and the range is the set of all second elements of these pairs (do not double these elements in the set): R = {0, 1} Exercise 4. Consider the following (infinite) set: F = {(p, 1), p Z, if p is even} {(p, 0), p Z, if p is odd} 1. Is this a valid representation of a function? 2. If yes, what are its range and domain? 4 Operations on functions Short introduction to the algebraic properties of the set of function. Main part: Composite functions 5 Properties 5.1 Is that odd? 5.2 Translation Along the x and afterwards along the y axis. 5.3 Reflection 5.4 Invertability Horizontal line test Similarity about y = x 5.5 Opening Translating and reflecting a graph is equivalent to composing a function with a linear one (either to the left or to the right). Of interest for the next chapter! Exercise 5. Effect of adding a slope to that linear function we are composing with? Want to talk about piece-wise functions? Constant functions? 7