1. 1 Copyright © Cengage Learning. All rights reserved. 3 Functions and Graphs 3.4 Definition of function

2. 2 Definition of Function The notion of correspondence occurs frequently in everyday life. Some examples are given in the following illustration. Illustration: Correspondence To each book in a library there corresponds the number of pages in the book. To each human being there corresponds a birth date. If the temperature of the air is recorded throughout the day, then to each instant of time there corresponds a temperature.

3. 3 Definition of Function The element x of D is the argument of f. The set D is the domain of the function. The element y of E is the value of f at x (or the image of x under f ) and is denoted by f (x), read “f of x.” The range of f is the subset R of E consisting of all possible values f (x) for x in D. Note that there may be elements in the set E that are not in the range R of f.

4. 4 Definition of Function Consider the diagram in Figure 2. The curved arrows indicate that the elements f (w), f (z), f (x) and f (a) of E correspond to the elements w, z, x, and a of D. Figure 2

5. 5 Definition of Function To each element in D there is assigned exactly one function value in E; however, different elements of D, such as w and z in Figure 2, may have the same value in E. The symbols f : D  E, and signify that f is a function from D to E, and we say that f maps D into E. Initially, the notations f and f (x) may be confusing. Remember that f is used to represent the function.

6. 6 Definition of Function It is neither in D nor in E. However f (x), is an element of the range R—the element that the function f assigns to the element x, which is in the domain D. Two functions f and g from D to E are equal, and we write f = g provided f (x) = g(x) for every x in D. For example, if g(x) = (2x 2 – 6) + 3 and f (x) = x 2 for every x in, then g = f.

7. 7 Example 1 – Finding function values Let f be the function with domain such that f (x) = x 2 for every x in. (a) Find f (– 6), f ( ), f (a + b), and f (a) + f (b), where a and b are real numbers. (b) What is the range of f ? Solution: (a) We find values of f by substituting for x in the equation f (x) = x 2 : f (– 6) = (– 6) 2 = 36 f ( ) = ( ) 2 = 3

8. 8 Example 1 – Solution f (a + b) = (a + b) 2 = a 2 + 2ab + b 2 f (a) + f (b) = a 2 + b 2 (b) By definition, the range of f consists of all numbers of the form f (x) = x 2 for x in. Since the square of every real number is nonnegative, the range is contained in the set of all nonnegative real numbers. Moreover, every nonnegative real number c is a value of f, since f ( ) = ( ) 2 = c. Hence, the range of f is the set of all nonnegative real numbers. cont’d

9. 9 Definition of Function If a function is defined as in Example 1, the symbols used for the function and variable are immaterial; that is, expressions such as f (x) = x 2, f (s) = s 2, g (t) = t 2, and k(r) = r 2 all define the same function. This is true because if a is any number in the domain, then the same value a 2 is obtained regardless of which expression is employed. The phrase f is a function will mean that the domain and range are sets of real numbers.

10. 10 Definition of Function If a function is defined by means of an expression, as in Example 1, and the domain D is not stated, then we will consider D to be the totality of real numbers x such that f (x) is real. This is sometimes called the implied domain of f. To illustrate, if f (x) = then the implied domain is the set of real numbers x such that is real—that is, x – 2  0, or x  2. Thus, the domain is the infinite interval [2, ).

11. 11 Definition of Function If x is in the domain, we say that f is defined at x or that f (x) exists. If a set S is contained in the domain, f is defined on S. The terminology f is undefined at x means that x is not in the domain of f.

12. 12 Example 2 – Finding function values Let g (x) = (a) Find the domain of g. (b) Find g(5), g(–2), g(–a), and –g(a). Solution: (a) The expression is a real number if and only if the radicand 4 + x is nonnegative and the denominator 1 – x is not equal to 0. Thus, exists if and only if 4 + x  0 and 1 – x ≠ 0

13. 13 Example 2 – Solution or, equivalently, x  – 4 and x ≠ 1. We may express the domain in terms of intervals as [– 4, 1)  (1, ). (b) To find values of g, we substitute for x: cont’d

14. 14 Example 2 – Solution cont’d

15. 15 Definition of Function Graphs are often used to describe the variation of physical quantities. For example, a scientist may use the graph in Figure 5 to indicate the temperature T of a certain solution at various times t during an experiment. Figure 5

16. 16 Definition of Function The sketch shows that the temperature increased gradually from time t = 0 to time t = 5, did not change between t = 5 and t = 8, and then decreased rapidly from t = 8 to t = 9. Similarly, if f is a function, we may use a graph to indicate the change in f (x) as x varies through the domain of f. Specifically, we have the following definition.

17. 17 Definition of Function In general, we may use the following graphical test to determine whether a graph is the graph of a function. The x-intercepts of the graph of a function f are the solutions of the equation f (x) = 0. These numbers are called the zeros of the function. The y-intercept of the graph is f (0), if it exists.

18. 18 Example 3 – Sketching the graph of a function Let f (x) = (a) Sketch the graph of f. (b) Find the domain and range of f. Solution: (a) By definition, the graph of f is the graph of the equation y = The following table lists coordinates of several points on the graph.

19. 19 Example 3 – Solution Plotting points, we obtain the sketch shown in Figure 7. Note that the x-intercept is 1 and there is no y-intercept. cont’d Figure 7

20. 20 Example 3 – Solution (b) Referring to Figure 7, note that the domain of f consists of all real numbers x such that x  1 or, equivalently, the interval [1, ). The range of f is the set of all real numbers y such that y  0 or, equivalently, [0, ). cont’d

21. 21 Definition of Function The square root function, defined by f (x) = has a graph similar to the one in Figure 7, but the endpoint is at (0, 0). Figure 7

22. 22 Definition of Function The y-value of a point on this graph is the number displayed on a calculator when a square root is requested. This graphical relationship may help you remember that is 3 and that is not  3. Similarly, f (x) = x 2, f (x) = x 3, and f (x) = are often referred to as the squaring function, the cubing function, and the cube root function, respectively.

23. 23 Definition of Function In general, we shall consider functions that increase or decrease on an interval I, as described in the following chart, where x 1 and x 2 denote numbers in I. Increasing, Decreasing, and Constant Functions

24. 24 Definition of Function cont’d

25. 25 An example of an increasing function is the identity function, whose equation is f (x) = x and whose graph is the line through the origin with slope 1. An example of a decreasing function is f (x) = – x, an equation of the line through the origin with slope –1. If f (x) = c for every real number x, then f is called a constant function. We shall use the phrases f is increasing and f (x) is increasing interchangeably. We shall do the same with the terms decreasing and constant. Definition of Function

26. 26 Example 4 – Using a graph to find domain, range, and where a function increases or decreases Let f (x) = (a) Sketch the graph of f. (b) Find the domain and range of f. (c) Find the intervals on which f is increasing or is decreasing. Solution: (a) By definition, the graph of f is the graph of the equation y = We know from our work with circles that the graph of x 2 + y 2 = 9 is a circle of radius 3 with center at the origin.

27. 27 Example 4 – Solution Solving the equation x 2 + y 2 = 9 for y gives us y =  It follows that the graph of f is the upper half of the circle, as illustrated in Figure 8. cont’d Figure 8

28. 28 Example 4 – Solution (b) Referring to Figure 8, we see that the domain of f is the closed interval [–3, 3], and the range of f is the interval [0, 3]. (c) The graph rises as x increases from –3 to 0, so f is increasing on the closed interval [–3, 0]. Thus, as shown in the preceding chart, if x 1 < x 2 in [–3, 0], then f (x 1 ) < f (x 2 ) (note that possibly x 1 = –3 or x 2 = 0) cont’d

29. 29 Example 4 – Solution The graph falls as x increases from 0 to 3, so f is decreasing on the closed interval [0, 3]. In this case, the chart indicates that if x 1 f (x 2 ) (note that possibly x 1 = 0 or x 2 = 3) cont’d

30. 30 Of special interest in calculus is a problem of the following type. Problem: Find the slope of the secant line through the points P and Q shown in Figure 9. Definition of Function Figure 9

31. 31 The slope is m PQ given by m PQ The last expression (with h ≠ 0) is commonly called a difference quotient. Definition of Function

32. 32 Example 5 – Simplifying a difference quotient Simplify the difference quotient using the function f (x) = x 2 + 6x – 4. Solution: definition of f expand numerator

33. 33 = 2x + h + 6 Example 5 – Solution subtract terms simplify factor out h cancel h ≠ 0 cont’d

34. 34 The following type of function is one of the most basic in algebra. The graph of f in the preceding definition is the graph of y = ax + b, which, by the slope-intercept form, is a line with slope a and y-intercept b. Thus, the graph of a linear function is a line. Definition of Function

35. 35 Since f (x) exists for every x, the domain of f is As illustrated in the next example, if a ≠ 0 then the range of f is also Definition of Function

36. 36 Example 6 – Sketching the graph of a linear function Let f (x) = 2x + 3. (a) Sketch the graph of f. (b) Find the domain and range of f. (c) Determine where f is increasing or is decreasing. Solution: (a) Since f (x) has the form ax + b, with a = 2 and b = 3, f is a linear function.

37. 37 Example 6 – Solution The graph of y = 2x + 3 is the line with slope 2 and y-intercept 3, illustrated in Figure 10. cont’d Figure 10

38. 38 Example 6 – Solution (b) We see from the graph that x and y may be any real numbers, so both the domain and the range of f are (c) Since the slope a is positive, the graph of f rises as x increases; that is, f (x 1 ) < f (x 2 ) whenever x 1 < x 2. Thus, f is increasing throughout its domain. cont’d

39. 39 Many formulas that occur in mathematics and the sciences determine functions. For instance, the formula A =  r 2 for the area A of a circle of radius r assigns to each positive real number r exactly one value of A. This determines a function f such that f (r) =  r 2, and we may write A = f (r). The letter r, which represents an arbitrary number from the domain of f, is called an independent variable. Definition of Function

40. 40 The letter A, which represents a number from the range of f, is a dependent variable, since its value depends on the number assigned to r. If two variables r and A are related in this manner, we say that A is a function of r. In applications, the independent variable and dependent variable are sometimes referred to as the input variable and output variable, respectively. Definition of Function

41. 41 As another example, if an automobile travels at a uniform rate of 50 mi/hr, then the distance d (miles) traveled in time t (hours) is given by d = 50t, and hence the distance d is a function of time t. Definition of Function

42. 42 Example 8 – Expressing the volume of a tank as a function of its radius A steel storage tank for propane gas is to be constructed in the shape of a right circular cylinder of altitude 10 feet with a hemisphere attached to each end. The radius r is yet to be determined. Express the volume V (in ft 3 ) of the tank as a function of r (in feet). Solution: The tank is illustrated in Figure 12. Figure 12

43. 43 Example 8 – Solution We may find the volume of the cylindrical part of the tank by multiplying the altitude 10 by the area  r 2 of the base of the cylinder. This gives us volume of cylinder = 10(  r 2 ) = 10  r 2. The two hemispherical ends, taken together, form a sphere of radius r. Using the formula for the volume of a sphere, we obtain volume of the two ends =  r 3. cont’d

44. 44 Example 8 – Solution Thus, the volume V of the tank is V = + 10  r 2 This formula expresses V as a function of r. In factored form, V(r) = (4r + 30) = (2r + 15) cont’d

45. 45 Ordered pairs can be used to obtain an alternative approach to functions. We first observe that a function f from D to E determines the following set W of ordered pairs: W = {(x, f (x)): x is in D} Thus, W consists of all ordered pairs such that the first number x is in D and the second number is the function value f (x). In Example 1, where f (x) = x 2, W is the set of all ordered pairs of the form (x, x 2 ). Definition of Function

46. 46 It is important to note that, for each x, there is exactly one ordered pair (x, y) in W having x in the first position. Conversely, if we begin with a set W of ordered pairs such that each x in D appears exactly once in the first position of an ordered pair, then W determines a function. Specifically, for each x in D there is exactly one pair (x, y) in W, and by letting y correspond to x, we obtain a function with domain D. Definition of Function

47. 47 The range consists of all real numbers y that appear in the second position of the ordered pairs. It follows from the preceding discussion that the next statement could also be used as a definition of function. Definition of Function