1.2 Functions What is a Function? 1.2. FUNCTIONS 11

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1 1.2. FUNCTIONS Functions What is a Function? In this section, we only consider functions of one variable. Loosely speaking, a function is a special relation which exists between two variables. In introductory mathematics classes, the definition below is the one which is usually given for a function. Definition 22 (function) Let A and B denote two sets. 1. A function f from A to B is a rule which assigns a unique y B to each x A. We write y = f (x). f (x) denotes the value of the function at x. 2. x is called the independent variable (also called an input value), y is the dependent variable (also called an output value). Remember, if we say that y is a function of x, it implies that it depends on x. 3. The domain of f is A, the set of values of x. It is also denoted D (f) or Dom f. When the domain of a function is not given, it is understood to be the largest set of real numbers for which the function is defined. 4. The range of f is the set R (f) = {f (x) : x D (f)}. It is also denoted Range f. A function can also be defined in terms of pairs. In this manner, we simply give the pairs of elements which are in relation. More precisely, Definition 23 (function) Let A and B denote two sets. A function from A to B is a subset of A B that is a set of ordered pairs with the property that whenever (a, b) f and (a, c) f then b = c. When we say that (a, b) f, it means that b = f (a). With this form of the definition, the domain of f is simply the set of first members of each pair, the range is the set of second members. A set of ordered pairs is a function if no two pairs have the same first member and different second member. At this stage, it is also important for students to understand the difference between f and f (x). f is a function. As we have seen in the above definition, it is a set of ordered pairs. f (x) is an element of R (f), the range of f. A function from A into R is called real-valued. Definition 24 (mapping) If f is a function from A to B, we also say that f is a mapping from A into B, or that f maps A into B. We often write: f : A B Definition 25 (image) If f is a function from A to B and y = f (x) (or if (x, y) f), then we say that y is the image of x under f. The following proposition follows from the definition of a function.

2 12CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU Proposition 26 (vertical line test) A graph is the graph of a function if it passes the vertical line test, that is if no vertical line can intersect the graph in more than one point. Example 27 Let A = {0, 1, 2, 3, 4, 5} and B = R. Let f = {(0, 6), (1, 54), (2, 70), (3, 54), (4, 6), (5, 74)} It is easy to see that f is a function. Each x A belongs to exactly one pair in f. The fact that some pairs have the same second component does not contradict the definition of a function. Though it indicates something else we will study soon. Example 28 With A and B as in the previous example, g = {(0, 6), (1, 54), (1, 74), (2, 70), (3, 54), (4, 6), (5, 74)} is not a function. The pairs (1, 54) and (1, 74) have the same first component but have diff erent second components. This violates the definition of a function. Example 29 Often, a function is given by a formula which gives the relationship between input and output values as in this example. Let A = B = R. We define the function h by h = { (x, y) R 2 : y = x } In other words, h (x) = x Example 30 The identity function, denoted i, is defined on any non-empty set A by: i = {(x, x) : x A} In other words, this function maps every element of A into itself, that is i (x) = x for any x A. Remark 31 A formula is not enough to define a function. Its domain must also be specified. Two functions are equal if they have the same set of ordered pairs. When functions are given by a formula, two functions are equal if they have the same domain and they define the same relation between input and output values Restrictions and Extensions of Functions Consider a function f from A to B. Let D 1 A. We can define a new function f 1 with domain D 1 by : f 1 (x) = f (x) x D 1 Remark 32 The quantifier means "for every".

3 1.2. FUNCTIONS 13 Definition 33 (restriction) The function f 1 as defined above is called a restriction of f to D 1. It is called a restriction because it is defined on a smaller set. On the smaller sets, the two functions agree. In other words, f 1 is the same as f, restricted to D 1. Example 34 The function g : [ π 2, π ] [ 1, 1] 2 x sin x is a restriction of the function to the set [ π 2, π ]. 2 f : R [ 1, 1] x sin x Remark 35 We restrict functions to a smaller domain when we wish the function to have certain properties it does not have on the larger domain. In the example above, you will recall from calculus that though sin x is defined for all real numbers, it is not invertible because[ it does not pass the horizontal line test. However, if we restrict its domain to π 2, π ], the function is invertible and 2 we haven t lost any information because on that interval, sin x takes on all its values. You may recall that sin x is periodic [ (see definition below if you have forgotten what it means). The interval π 2, π ] corresponds to one period. 2 Instead of restricting the definition of a function to a smaller set, we can also do the opposite. Sometimes, it is useful to extend the definition of a function to a larger set. Let g be a function from A into B, and let D 2 be a set containing A. We can define a new function g 2 with domain D 2 by: g 2 (x) = g (x) x A g 2 (x) = some other function if x D 2 \ A Definition 36 The function g 2 as defined above is called an extension of g to D 2. It is called an extension because we extend the definition of g to a larger set. Example 37 The function h : R R { 1 if x = 0 x sin x x if x 0

4 14CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU is an extension of the function f : R\ {0} R x sin x x to the set of all real numbers. In this case, the reason for doing this is to sin x have a function continuous on the set of real numbers. is continuous on x R\{0} because it is not defined at 0. However, you may recall from calculus that sin x lim = 1. We can extend sin x to a continuous function on R with the x 0 x x function of this example Operations on Functions Addition, Subtraction, Multiplication and Division Usually, a function is defined by defining the action this function has on elements of its domain. Let f and g be two functions, call D (f) the domain of f and D (g) the domain of g. We define addition, subtraction, multiplication and division of functions as follows: The sum of two functions f and g, denoted f + g, is defined by: (f + g) (x) = f (x) + g (x) The domain of f + g is the set {x : x D (f) and x D (g)}. The difference of two functions f and g, denoted f g, is defined by: (f g) (x) = f (x) g (x) The domain of f g is the set {x : x D (f) and x D (g)}. The product of two functions f and g, denoted fg, is defined by: (fg) (x) = f (x) g (x) The domain of fg is the set {x : x D (f) and x D (g)}. The division of two functions f and g, denoted f, is defined by: g f f (x) (x) = g g (x) The domain of f is the set {x : x D (f) and x D (g) with g (x) 0}. g

5 1.2. FUNCTIONS 15 Example 38 Given f (x) = x + 1 and g (x) = x 1, find f + g, f g, fg, f g, find their domain. We find these functions by defining how they act on elements. Since both f and g are polynomials, D (f) and D (g) are the set of real numbers. (f + g) (x) = f (x) + g (x) = x x 1 = 2x. The domain of f + g is the set of real numbers. (f g) (x) = f (x) g (x) = x + 1 (x 1) = 2. The domain of f g is the set of real numbers. (fg) (x) = f (x) g (x) = (x + 1) (x 1) = x 2 1. The domain of fg is the set of real numbers. f f (x) (x) = g g (x) = x + 1 x 1. The domain of f g 1. Composition is the set of real numbers except Once again, let f be a function from A into B and g be a function from B into C. The composition of f and g, denoted g f, is the function defined by: (g f) (x) = g (f (x)) We can also define the composition of two functions in terms of ordered pairs. let f be a function from A into B and g be a function from B into C. Then, g f = {(a, c) A C : b B for which (a, b) f and (b, c) g} = {(a, c) A C : c = g (f (a))} Remark 39 The quantifier means "there exists". The domain of g f is the set {x D (f) : f (x) D (g)} that is it is the set of elements in the domain of f such that f (x) is in the domain of g. Example 40 Let f (x) = x and g (x) = x Find f g, g f and their domain. We find f g by finding how it acts on an element x. By definition, we have: (f g) (x) = f (g (x)) = f ( x ) x2 + 1 Since x is always defined, and always positive, the domain of f g is all real numbers.

6 16CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU We find g f by finding how it acts on an element x. By definition, we have: (g f) (x) = g (f (x)) = g ( x ) = ( x ) = x + 1 The domain of g f is {x R : x 0}. Remark 41 You will notice that in general f g g f Example 42 Find functions f and g such that (f g) (x) = x 1 It is important to notice that when we write (f g) (x), the function g is applied first, then the function f. When we write x 1, first, we have to evaluate x 1, then we take the square root. This suggests that g (x) = x 1, and f (x) = x. If we verify, we get (f g) (x) = f (g (x)) = f (x 1) x 1 Remark 43 In general, the notation f n is used to denote composition of f with itself n times. (f f) (x) = f (f (x)) = f 2 (x). However, with trigonometric functions, raising to a power means multiplication. For example sin 2 x = (sin x) 2 = (sin x) (sin x). Do not confuse this with f (n) which means n th order derivative Injections and Surjections Definition 44 (surjection) If f is a mapping of A into B such that R (f) = B, then we say that the mapping is onto. We also say that f is surjective, or that f is a surjection. To put it simply, a mapping from A into B is a surjection if every element of B is the image of some element of A. To prove that a function f from A to B is a surjection, one must prove that for any y B, there exists an x in A such that y = f (x). Remark 45 A function f : A R (f) is always a surjection. Example 46 The function f : R R x x 2

7 1.2. FUNCTIONS 17 is not surjective. However, the function is surjective. f : R [0, ) x x 2 Definition 47 (injection) Let f be a function from A into B. f is said to be injective, or an injection, or one-to-one if one of the three equivalent conditions below is satisfied. 1. f (a) = f (b) a = b 2. a b f (a) f (b) 3. (a, c) f and (b, c) f a = b To put it simply, a mapping from A into B is an injection if different inputs produce different outputs. Example 48 The function f : R [ 1, 1] x sin x is [ not an injection, f (0) = f (2π) for example. π 2, π ] is an injection. 2 However, its restriction to Proposition 49 (horizontal line test) The graph of a function is the graph of an injective function if it passes the horizontal line test, that is if no horizontal line can intersect the graph in more than one point. Definition 50 A function which is both an injection (one-to-one) and a surjection (onto) is called a bijection Inverse Functions Proposition 51 Let f be a function from A onto B. If f is an injection, then the function f 1 : B A such that f 1 = {(b, a) B A : (a, b) f} is also a function which is one-to-one. Proof. The proof of this fact is left as an exercise. The function f 1 defined in terms of f has a name: Definition 52 (inverse) Let f be an injective function from A onto B. The function f 1 = {(b, a) B A : (a, b) f} is called the inverse of f. It is denoted f 1.

8 18CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU f and f 1 are related in many different ways. We list a few of these relations below. 1. D (f) = R ( f 1), R (f) = D ( f 1). This can be seen from the definition. 2. y = f (x) x = f 1 (y). This can be seen from the definition. 3. ( f 1 f ) (x) = x x D (f). See problems at the end of this section. 4. ( f f 1) (y) = y y R (f). See problems at the end of this section. You may recall from previous mathematics classes that to find the inverse of a function given by a formula y = f (x), the following steps can be followed: 1. Make sure the function has an inverse (i.e. it is one-to-one). 2. In the relation y = f (x), switch x and y. 3. Solve for y in the relation you obtained above. 4. The new relation you obtained for y is the inverse function. Example 53 Find the inverse of y = f (x) = 5x + 2. This is a straight line which is not horizontal, so it passes the horizontal line test. Hence it is one-to-one. If we switch x and y, we obtain Next, we solve for y. x = 5y + 2 Therefore x = 5y + 2 x 2 = 5y y = x 2 5 f 1 (x) = x Direct Image and Inverse Image of a Set Let f be a function from A into B. Let E and G be two sets such that E A, and G B. Definition 54 (direct image of a set) The direct image of E, denoted f (E), is defined by: f (E) = {f (x) : x E} Definition 55 (inverse image of a set) The inverse image of G, denoted f 1 (G) is defined by: f 1 (G) = {x A : f (x) G}

9 1.2. FUNCTIONS 19 Remark 56 The above definition does not require that f be injective or have an inverse. f 1 (G) is simply the notation for the inverse image of G. The reader should never think we are talking about the inverse of f. Remark 57 If should be clear to the reader that f (E) B. Also f (A) is the range of f. Therefore, f (A) = B f is a surjection. Remark 58 Similarly, it should be clear to the reader that f 1 (G) A. Let us consider an example to illustrate this definition. Example 59 Consider f : N N defined by f (n) = n Let E = {2, 3, 4}. Find f (E). By definition, f (E) = {f (x) : x E} in other words, f (E) = {f (2), f (3), f (4)} = {3, 8, 15} 2. Let G = {3, 4, 5, 6, 7, 8}, find f 1 (G). By definition, f 1 (G) = {x N : f (x) G}. The only whole numbers mapping into an element of G are 2 and 3 since f (2) = 3 and f (3) = 8. Hence f 1 (G) = {2, 3} Certain important set properties are preserved under the direct image or the inverse image of a set. We list them in two theorems. Theorem 60 Let f be a function from A into B. Let E and F be subsets of A. The following is true: 1. If E F then f (E) f (F ) 2. f (E F ) f (E) f (F ) 3. f (E F ) = f (E) f (F ) 4. f (E \ F ) f (E) Proof. We only prove some of these items. For the remaining ones, see the problems at the end of this section. 1. Let y f (E). Then, there exists x E such that y = f (x). Because E F, x is also in F, therefore, y = f (x) is in f (F ). 2. see problems 3. We need to show the inclusion both ways.

10 20CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU First, we show that f (E F ) f (E) f (F ). Let y f (E F ). Then, there exists x E F such that y = f (x). Either x E, in which case y = f (x) f (E). Or, x F, in which case y = f (x) f (F ). Thus, y f (E) f (F ) Next, we show that f (E) f (F ) f (E F ). Since E E F, by part 1, f (E) f (E F ). Similarly, F E F thus f (F ) f (E F ). It follows that f (E) f (F ) f (E F ). 4. Let y f (E \ F ). Then, there exists x E \F such that y = f (x). Then, x E, thus y = f (x) f (E). Remark 61 To prove part 4 of the above theorem, we can also use part 1 and the fact that E \ F E. Theorem 62 Let f be a function with domain in A and range in B. Let G and H be subsets of B. The following is true: 1. If G H then f 1 (G) f 1 (H) 2. f 1 (G H) = f 1 (G) f 1 (H) 3. f 1 (G H) = f 1 (G) f 1 (H) 4. f 1 (G \ H) = f 1 (G) \ f 1 (H) Proof. We only prove some of these items. For the remaining ones, see the problems at the end of this section. 1. see problems 2. We need to show the inclusion both ways. First, we show that f 1 (G H) f 1 (G) f 1 (H). Since G H G, by part 1, f 1 (G H) f 1 (G). Similarly, f 1 (G H) f 1 (H). Thus, f 1 (G H) f 1 (G) f 1 (H). Next, we show that f 1 (G) f 1 (H) f 1 (G H). Let x f 1 (G) f 1 (H). Then, f (x) G and f (x) H. Therefore, f (x) G H hence, x f 1 (G H) by definition. 3. see problems 4. see problems

11 1.2. FUNCTIONS Additional Properties of a Function General Properties In this section, we remind the reader of some definitions. Theorems about these properties will be proven later in the chapter. Definition 63 Let f : D R be a real-valued function. f is said to be: 1. Increasing, if a, b D (a b = f (a) f (b)) 2. Decreasing, if a, b D (a b = f (a) f (b)) 3. Strictly increasing, if a, b D (a < b = f (a) < f (b)) 4. Strictly decreasing, if a, b D (a < b = f (a) > f (b)) 5. Monotone, if it is either increasing or decreasing. 6. Strictly monotone, if it is either strictly increasing or strictly decreasing. 7. Bounded above, if its range is bounded above, that is if M R : x D f (x) M 8. Bounded below, if its range is bounded below, that is if m R : x D f (x) m 9. Bounded, if it is both bounded above and below. 10. Even, if x D x D and f ( x) = f (x) 11. Odd, if x D x D and f ( x) = f (x) { x + T D 12. Periodic, if T 0 R : x D and f (x + T ) = x T D f (x). The smallest such T is called the period of the function. 13. Lipschitz, if k > 0 R : a, b D f (a) f (b) k a b Global versus Local Properties Let f : D R be a real-valued function. A certain property is said to be a global property if it is true wherever the function is defined. However, certain properties are only true on an interval of the domain of the function. Such properties are called local properties. More generally, if f : D R is a realvalued function and E D, we say that f has a certain property on E if the restriction of f on E has this property. For example, the function f (x) = 1 is not decreasing on its domain which x is R \ {0} (why?). However, its restriction to (0, ) is decreasing, so is its restriction to (, 0).

12 22CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU If a D, we say that f has a certain property in a neighborhood of a if there exists an open set U containing a such that f has the property on U D. For example, the function f (x) = sin x is increasing in a neighborhood of 0; it is increasing on Exercises ( π 2, π 2 ). 1. Is the subset { (x, y) : x R,y R, and x 2 + y 2 = 1 } a function? Explain. 2. Define the function f and g by f (x) = x2 1 x 1 g (x) = x + 1 for x R. Is f = g? Why? for x R and x 1 and 3. Prove that if f is an injection from A to B, then f 1 = {(b, a) : (a, b) f} is a function. Then, prove this function is an injection. 4. Suppose that f is an injection. Show that ( f 1 f ) (x) = x for every x in D (f). Also, show that f f 1 (y) = y for every y in R (f). 5. Let f be a function from A into B. Let E and F be subsets of A. (a) Prove that f (E F ) f (E) f (F ). (b) Give an example which shows why the two sets are not equal. (c) When do you think the two sets are equal, why? 6. Let f be a function with domain in A and range in B. Let G and H be subsets of B. (a) Prove that if G H then f 1 (G) f 1 (H) (b) Prove that f 1 (G H) = f 1 (G) f 1 (H) (c) Prove that f 1 (G H) = f 1 (G) f 1 (H) 7. Give at least one example of a function for each of the 13 definition in definition 63. This means you will give at least 13 examples since there are 13 concepts defined. In each case, you will specify the domain of the function and you will prove that the function satisfies the property you claim it satisfies. 8. Let A = { 1, 0, 1, 2} and B = N. Which of the following subsets of A B are functions from A into B, explain. (a) f = {( 1, 2), (0, 3), (2, 5)} (b) g = {( 1, 2), (0, 7), (1, 1), (1, 3), (2, 7)} (c) h = {( 1, 2), (0, 2), (1, 2), (2, 1)} (d) k = {(x, y) : y = 2x + 3, x A}

13 1.2. FUNCTIONS Let f : N N be the function defined by f (n) = 2n 1. Find f (E) and f 1 (E) for each of the following subsets E of N. (a) {1, 2, 3, 4} (b) {1, 3, 5, 7} (c) N 10. Let f = { (x, y) : x R, y = x }. (a) Let A = {x : 1 x 2}. Find f (A) and f 1 (A). (b) Show that f is an injection and a surjection. (c) Find f For each of the following real-valued functions, find the range of the functions f and determine if the function is one-to-one. If f is one-to-one, find the inverse function f 1 and specify the domain of f 1. (a) f (x) = 3x 2, D (f) = R. (b) f (x) = sin x, D (f) = {x R : 0 x π}.