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Unformatted text preview: Unit 3 Functions Introduction Introduction In mathematics you often work with situations in which one quantity depends on another. For example: • The distance walked by a woman at a particular speed depends on the time that she’s been walking. • The height of a gondola on a Ferris wheel depends on the angle through which the wheel has rotated since the gondola was in its lowest position. • The number of 5-litre tins of a particular type of paint needed by a decorator depends on the area that he intends to paint. Whenever one quantity depends on another, we say that the first quantity is a function of the second quantity. The idea of a function is fundamental in mathematics, and in particular it forms the foundation for calculus, which you’ll begin to study in Unit 6. In this unit you’ll be introduced to the terminology and notation that are used for functions. You’ll learn about some standard, frequently-arising types of functions, and how to use graphs to visualise properties of functions. You’ll also learn how you can use your knowledge about a few standard functions to help you understand and work with a wide range of related functions. Later in the unit you’ll revise exponential functions and logarithms, and practise working with them. In the final section you’ll revise inequalities, and see how working with functions and their graphs can help you understand and solve some quite complicated inequalities. A Ferris wheel This is a long unit. The study calendar allows extra time for you to study it. 1 Functions and their graphs This section introduces you to the idea of a function and its graph, and shows you some standard functions. You’ll start by learning about sets, which are needed when you work with functions and also in many other areas of mathematics. 201 Unit 3 Functions 1.1 Sets of real numbers In mathematics a set is a collection of objects. The objects could be anything at all: they could be numbers, points in the plane, equations or anything else. For example, each of the following collections of objects forms a set: • all the prime numbers less than 100 • all the points on any particular line in the plane • all the equations that represent vertical lines • the solutions of any particular quadratic equation. A set can contain any number of objects. It could contain one object, two objects, twenty objects, infinitely many objects, or even no objects at all. Each object in a set is called an element or member of the set, and we say that the elements of the set belong to or are in the set. There are many ways to specify a set. If there are just a few elements, then you can list them, enclosing them in curly brackets. For example, you can specify a set S as follows: S = {3, 7, 9, 42}. Another simple way to specify a set is to describe it. For example, you can say ‘let T be the set of all even integers’ or ‘let U be the set of all real numbers greater than 5’. We usually denote sets by capital letters. The set that contains no elements at all is called the empty set, and is denoted by the symbol ∅. It’s often useful to state that a particular object is or is not a member of a particular set. You can do this concisely using the symbols ∈ and .∈, which mean ‘is in’ and ‘is not in’, respectively. For example, if S is the set specified above, then the following statements are true: 7∈S and Activity 1 10 .∈ S. Understanding set notation Let X = {1, 2, 3, 4} and let Y be the set of all odd integers. Which of the following statements are true? (a) 1 ∈ X (b) 1 ∈ Y (c) 2 .∈ X (d) 2 .∈ Y It’s often useful to construct ‘new sets out of old sets’. For example, if A and B are any two sets, then you can form a new set whose members are all the objects that belong to both A and B. This set is called the intersection of A and B, and is denoted by A ∩ B. For instance, if A = {1, 2, 3, 4} 202 and B = {3, 4, 5}, 1 Functions and their graphs then A ∩ B = {3, 4}. Similarly, if A and B are any two sets, then you can form a new set whose members are all the objects that belong to either A or B (or both). This set is called the union of A and B, and is denoted by A ∪ B. For example, if A and B are as specified above, then A ∪ B = {1, 2, 3, 4, 5}. You might find it helpful to visualise intersections and unions of sets by using diagrams like those in Figure 1, which are known as Venn diagrams. The Venn diagrams in the figure show the intersection and union of the particular sets A and B above. A B 1 3 2 4 (a) A 5 B 1 3 2 4 5 (b) Figure 1 (a) The intersection (shaded) and (b) the union (shaded) of two sets Venn diagrams are named after the logician John Venn, who used them in publications starting in 1880. However, the idea of using diagrams in this way did not originate with Venn. The prolific Swiss mathematician Leonhard Euler (pronounced ‘oiler’) used them in his Letters to a German Princess (1760–62). Venn acknowledged Euler’s influence by calling his own diagrams ‘Eulerian circles’. He extended Euler’s idea, using the diagrams to analyse more complex logical problems. As well as working on logic at Cambridge University, Venn was for some time a priest and later a historian. There is more about Euler on page 214. John Venn (1834–1923) You can form intersections and unions of more than two sets in a similar way. In general, the intersection of two or more sets is the set of all objects that belong to all of the original sets, and the union of two or more sets is the set of all objects that belong to any of the original sets. For example, if A and B are as specified above and C = {20, 21}, then A∩B∩C =∅ and A ∪ B ∪ C = {1, 2, 3, 4, 5, 20, 21}. 203 Unit 3 Functions Activity 2 Understanding unions and intersections of sets Let P = {1, 2, 3, 4, 5, 6}, let Q = {2, 4, 6, 8, 10, 12} and let R be the set of all integers divisible by 3. Specify each of the following sets. (a) P ∩ Q (b) Q ∩ R (c) P ∩ Q ∩ R (d) P ∪ Q The set membership symbol ∈ is a stylised version of the Greek letter ε (epsilon). The Italian mathematician Giuseppe Peano (1858–1932), the founder of symbolic logic, used ε to indicate set membership in a text published in 1889. He stated that it was an abbreviation for the Latin word ‘est’, which means ‘is’. The symbol was then adopted by the logician Bertrand Russell (1872–1970) in a text published in 1903, but it was typeset in a form that looks like the modern symbol ∈, and this form has remained in use to the present day. Peano also introduced the symbols ∩ and ∪ for intersection and union. The empty set symbol ∅ was introduced in 1939 by the influential French mathematician Andr´e Weil (1906–1998). It was inspired by the letter ø in the Norwegian alphabet. Sometimes every element of a set A is also an element of a set B. In this case we say that A is a subset of B, and we write A ⊆ B. For example: 1 3 2 • {1, 3} is a subset of {1, 2, 3} (as shown in Figure 2) • the set of integers is a subset of the set of real numbers. Every set is a subset of itself, and the empty set is a subset of every set. In this module, and particularly in this unit, you’ll mostly be working with sets whose elements are real numbers. In the rest of this subsection, you’ll meet some useful ways to visualise and represent sets of this type. Figure 2 A subset of a set (shaded) The set of all real numbers is denoted by R. You can handwrite this as: You saw in Unit 1 that you can visualise the real numbers as points on an infinitely long straight line, called the number line or the real line. Part of the number line is shown in Figure 3. Although only the integers are marked in the diagram, every point on the line represents a real number. − 10 − 9 − 8 − 7 − 6 − 5 − 4 − 3 − 2 − 1 0 1 2 3 4 5 6 7 8 9 10 Figure 3 The number line 204 1 Functions and their graphs You can use the number line to visualise sets of real numbers. For example: • Figure 4(a) shows the set {−1, 0, 1}. • Figure 4(b) shows the set of real numbers that are greater than or equal to 2 and also less than or equal to 6. • Figure 4(c) shows the set of real numbers that are greater than −5. • Figure 4(d) shows the set of real numbers that are less than greater than or equal to 3. 1 2 or In these kinds of diagrams, a solid dot indicates a number that’s included in the set, and a hollow dot indicates a number that isn’t included. A heavy line that continues to the left or right end of the diagram indicates that the set extends indefinitely in that direction. −3−2−1 0 1 2 3 (a) −7−6−5−4−3−2−1 0 1 2 3 4 5 6 7 8 (b) −3−2−1 0 1 2 3 4 5 1 2 (c) (d) Figure 4 Sets of real numbers The sets in Figure 4(b) and (c) are examples of a special type of set of real numbers, called an interval. An interval is a set of real numbers that corresponds to a part of the number line that you can draw ‘without lifting your pen from the paper’. The sets in Figure 4(a) and (d) aren’t intervals, as they have ‘gaps’ in them. In fact, the set in Figure 4(a) is the union of three intervals (each containing a single number), and the set in Figure 4(d) is the union of two intervals. A number that lies at an end of an interval is called an endpoint of the interval. For example, the interval in Figure 4(b) has two endpoints, namely 2 and 6, and the interval in Figure 4(c) has one endpoint, namely −5. The whole set of real numbers, R, is an interval with no endpoints. An interval that includes all of its endpoints is said to be closed, and one that doesn’t include any of its endpoints is said to be open. For example, the interval in Figure 4(b) is closed (since it includes both its endpoints), and the one in Figure 4(c) is open (since it excludes its single endpoint). An interval that includes one endpoint and excludes another, such as the interval in Figure 5, is said to be half-open (or half-closed). Since the interval R has no endpoints, it’s both open and closed! This fact may seem strange at the moment, but it will make more sense if you go on to study pure mathematics at higher levels. −2−1 0 1 2 3 4 Figure 5 A half-open (or half-closed) interval 205 Unit 3 Functions Activity 3 Recognising intervals State whether each of the sets below is an interval. For each set that is an interval, state whether it’s open, closed or half-open. (a) −3−2−1 0 1 2 3 (c) −3−2−1 0 1 2 3 (e) −3−2−1 0 1 2 3 (g) −3−2−1 0 1 2 3 (b) −3−2−1 0 1 2 3 (d) −3−2−1 0 1 2 3 (f) −3−2−1 0 1 2 3 (h) −3−2−1 0 1 2 3 A convenient way to describe most intervals is to use inequality signs. These are listed below, with their meanings. (Note that some texts use slightly different inequality signs: ! and " instead of ≤ and ≥.) Inequality signs < ≤ > ≥ is is is is less than less than or equal to greater than greater than or equal to For example, the interval in Figure 6(a) is the set of real numbers x such that x > 2 (that is, such that x is greater than 2). Similarly, the interval in Figure 6(b) is the set of real numbers x such that x > 1 and x ≤ 4 (that is, such that x is greater than 1 and x is less than or equal to 4). We usually write this description slightly more concisely, as follows: the interval is the set of real numbers x such that 1 < x ≤ 4 (that is, such that 1 is less than x, which is less than or equal to 4). 206 1 0 1 2 3 4 5 6 7 (a) Functions and their graphs −1 0 1 2 3 4 5 6 (b) Figure 6 Intervals It might help you to remember the meanings of the inequality signs if you notice that when you use either of the signs < or >, the lesser quantity is on the smaller, pointed side of the sign. The same is true for the signs ≤ and ≥, except that one quantity is less than or equal to the other, rather than definitely less than it. The statement ‘x > 2’ is called an inequality. In general, an inequality is a mathematical statement that consists of two expressions with an inequality sign between them. A statement such as ‘1 < x ≤ 4’ is called a double inequality. The two inequality signs in a double inequality always point in the same direction as each other. Activity 4 Using inequality signs to describe intervals (a) Draw diagrams similar to those in Figure 6 to illustrate the intervals described by the following inequalities and double inequalities. (i) 0 < x < 1 (ii) −3 ≤ x < 2 (iii) x ≤ 5 (iv) x > 4 (b) For each of the following diagrams, write down an inequality or double inequality that describes the interval illustrated. (i) −3−2−1 0 1 2 3 4 5 6 (iii) −4−3−2−1 0 1 (v) −1 0 1 2 3 4 5 6 7 (ii) −5−4−3−2−1 0 (iv) −5−4−3−2−1 0 1 (vi) 2 3 4 5 6 7 8 Another useful way to describe intervals is to use interval notation. For example, the interval described by the double inequality 4 ≤ x < 7 is denoted in interval notation by [4, 7). The square bracket indicates an included endpoint, and the round bracket indicates an excluded one. An interval that extends indefinitely is denoted by using the symbol ∞ (which is read as ‘infinity’), or its ‘negative’, −∞ (which is read as ‘minus infinity’), in place of an endpoint. For example, the interval described by the inequality x ≥ 5 is denoted by [5, ∞), and the interval described by the 207 Unit 3 Functions inequality x < 6 is denoted by (−∞, 6). We always use a round bracket next to ∞ or −∞ in interval notation. Here’s a summary of the notation. Interval notation Open intervals (a, b) ❝ a ❝ a<x<b b ❝ a Closed intervals [a, b] ! a ! a≤x≤b b ! a (a, ∞) (−∞, b) x>a x<b [a, ∞) (−∞, b] x≥a x≤b (−∞, ∞) ❝ b R ! b (−∞, ∞) {a} R x=a ! Half-open (or half-closed) intervals [a, b) (a, b] ! a ❝ a≤x<b b ❝ a ! a<x≤b b Notice that you’ve now seen two different meanings for the notation (a, b), where a and b are real numbers. It can mean either an open interval, or a point in the coordinate plane. The meaning is usually clear from the context. Activity 5 Using interval notation Write each of the intervals below in interval notation. (a) −3−2−1 0 1 2 3 4 5 6 (c) −4−3−2−1 0 1 (e) −1 0 1 2 3 4 5 6 7 (b) −5−4−3−2−1 0 (d) −5−4−3−2−1 0 1 (f) 2 3 4 5 6 7 8 Sometimes you need to work with sets of real numbers that are unions of intervals, like those in Figure 7. 208 1 −2−1 0 1 2 3 4 5 (a) Functions and their graphs −1 0 1 2 3 4 5 6 7 8 9 (b) Figure 7 Two unions of intervals You can denote a union of intervals in interval notation by using the usual notation for intervals together with the union symbol ∪. For example, the sets in Figure 7 can be written as (−∞, 1] ∪ [2, 4) and [0, 3) ∪ (3, 5] ∪ [7, 8], respectively. Activity 6 Denoting unions of intervals For each of the following diagrams, write the set illustrated in interval notation. (a) (b) −8−7−6−5−4−3−2−1 0 1 2 3 0 1 2 3 4 5 6 7 (c) −3−2−1 0 1 2 3 It’s often useful to state that a particular number lies in, or doesn’t lie in, a particular interval or union of intervals. You can do this concisely using the symbols ∈ and .∈ in the usual way. For example, as illustrated in Figure 8, 1 ∈ [0, 4] and − 1 .∈ [0, 4]. In the next subsection you’ll begin your study of functions. −1 0 1 2 3 4 5 Figure 8 The interval [0, 4] 1.2 What is a function? As mentioned in the introduction to this unit, whenever one quantity depends on another, we say that the first quantity is a function of the second quantity. Here are some more examples. • If a car is driving along a straight road, then its displacement s (in km) from some reference point depends on the time t (in hours) that has elapsed since the start of its journey. So s is a function of t. 209 Unit 3 Functions • The formula C = 2πr expresses the circumference C of a circle in terms of its radius r (with both C and r measured in the same units). So the value of C depends on the value of r, and hence C is a function of r. • Figure 9 An electrocardiogram (each high peak in voltage corresponds to a heartbeat) The electrical voltage between two points on a person’s skin either side of his or her heart (which can be measured using electrodes) changes rhythmically with every heartbeat. So the voltage V (in volts, say) depends on the time t (in seconds, say) that has elapsed since some point in time, and hence V is a function of t. There’s no simple formula for the relationship between t and V , but it’s often displayed as an electrocardiogram (ECG), like the one in Figure 9. In each of these examples, there’s a rule that converts each value of one variable (such as t, in the car example) to a value of the other variable (such as s, in the car example). You can think of the rule as a kind of processor that takes input values and produces output values, as illustrated in Figure 10. input value processor output value Figure 10 A processor that takes input values and produces output values For instance, in the car example, an input value of 1.2 (a time, in hours) might be converted by the processor to an output value of 60 (a displacement, in kilometres). Similarly, in the circle example, an input value of 3 (a radius, in centimetres) would be converted by the processor to an output value of 2π × 3 = 6π (a circumference, in centimetres). Sometimes the rule associated with a function can be expressed using a formula, and sometimes it can’t. In each of the three examples in the list above, there’s also a set of allowed input values, and a set of values within which every output value lies. For instance, with the car example, if the journey lasts three hours, then the allowed input values are the real numbers between 0 and 3 inclusive (the possible elapsed times, in hours), and the output values lie in the set R of real numbers (they are displacements of the car from the reference point, in kilometres). A function is a mathematical object that describes a situation like those listed above. It’s defined as follows. 210 1 Functions and their graphs A function consists of: • a set of allowed input values, called the domain of the function • a set of values in which every output value lies, called the codomain of the function • a process, called the rule of the function, for converting each input value into exactly one output value. It’s often useful to denote a function by a letter. If a function is denoted by f , say, then for any input value x, the corresponding output value is denoted by f (x), which is read as ‘f of x’. For example, suppose that we denote the function associated with the car example by f . If the rule of this function converts the input value 1.2 to the output value 60, then we write f (1.2) = 60. Similarly, suppose that we denote the function associated with the circle example by g. The rule of this function converts the input value 3 to the output value 6π, so we write g(3) = 6π. This type of notation is known as function notation. One use of function notation is for specifying the rule of a function, when this can be done using a formula. For example, suppose that h is the function whose domain and codomain each consist of all the real numbers, and whose rule is ‘square the input number’. Then, for example, h(2) = 4, h(5) = 25 and h(−1) = 1, and the rule of h can be written as h(x) = x2 . Similarly, the rule of the function associated with the circle example can be written as g(r) = 2πr. When you write down the rule of a function, it doesn’t matter what letter you use to represent the input value. So the rule of the function h above could also be written as, for example, h(t) = t2 or h(u) = u2 . The variable used to denote the input value of a function is sometimes called the input variable. It’s traditional to use the letters f , g and h for functions, and the letters x, t and u for input variables. Although you can use any letters, these ones are often used in general discussions about functions. The most standard letters are f for a function and x for an input variable. 211 Unit 3 Functions Activity 7 Understanding function notation (a) Suppose that f is the function whose domain and codomain each consist of all the real numbers, and whose rule is f (t) = 4t. Write down the values of f (5) and f (−3). (b) Suppose that g is the function whose domain and codomain each consist of all the real numbers, and whose rule can be written in words as ‘multiply the input...
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