1. Copyright © Cengage Learning. All rights reserved. Graphs; Equations of Lines; Functions; Variation 3

2. Copyright © Cengage Learning. All rights reserved. Section 3.6 Functions

3. 3 Objectives 1. Find the domain and range of a set of ordered pairs. 2. Determine whether a given equation defines y to be a function of x. 3. Evaluate a function written in function notation. 4. Graph a function and determine its domain and range. 1 1 2 2 3 3 4 4

4. 4 Objectives 5. Determine whether a graph represents a function. 5 5

5. 5 Find the domain and range of a set of ordered pairs 1.

6. 6 Domain and Range of a Function The Table 3-3 shows the number of medals won by United States athletes during five Winter Olympics. We can display the data in the table as a set of ordered pairs, where the first component represents the year and the second component represents the number of medals won by U. S. athletes. {(1992, 11), (1994, 13), (1998, 13), (2002, 34), (2006, 25), (2010, 37)} Table 3-3

7. 7 Domain and Range of a Function Relation: set of ordered pairs {(1992, 11), (1994, 13), (1998, 13), (2002, 34), (2006, 25), (2010, 37)} Domain: set of all first components of a relation {1992, 1994, 1998, 2002, 2006, 2010} Range: set of all second components of a relation {11, 13, 34, 25, 37}

8. 8 Example Find the domain and range of the relation {(–2, –5), (4, 7), (8, 9)}. Solution: The domain is the set of first components of the ordered pairs: {–2, 4, 8} The range is the set of second components of the ordered pairs: {–5, 7, 9}

9. 9 Domain and Range of a Function When to each first component in a relation, there corresponds exactly one second component, the relation is called a function. Function A function is a set of ordered pairs (a relation) in which to each first component, there corresponds exactly one second component.

10. 10 Determine whether a given equation defines y to be a function of x 2.

11. 11 y as Function of x We have constructed the following table of ordered pairs for the equation y = 3x – 4 by substituting specific values for x and computing the corresponding values of y. Since the equation determines a set of ordered pairs, the equation determines a relation. Input values Output values

12. 12 y as Function of x From the table or equation, we can see that each input value x determines exactly one output value y. Because this is true, the equation also defines y to be a function of x. This leads to the following definition. y is a Function of x Any equation in x and y where each value of x (the input) determines exactly one value of y (the output) is called a function. In this case, we say that y is a function of x.

13. 13 y as Function of x The set of all input values x is called the domain of the function, and the set of all output values y is called the range of the function. Comment A function is always a relation, but a relation is not necessarily a function. For example, the relation {(2, 1), (2, 3)} is not a function because the input 2 determines two different values in the range: 1 and 3.

14. 14 y as Function of x Since each value of y in a function depends on a specific value of x, y is the dependent variable and x is the independent variable. The graph of a function is the graph of the equation that defines the function. In Example 2, we graph a linear equation. Any function that is defined by a linear equation is called a linear function.

15. 15 Example Determine whether the equations define y to be a function of x. a. y = x 2 b. x = y 2 Solution: a. We construct a table of ordered pairs for the equation y = x 2 by substituting values for x and computing the corresponding values.

16. 16 Example – Solution From the table we can see that each input value x determines exactly one output value y. The relation is a function. cont’d If x = –2, then y = (–2) 2 = 4. If x = 2, then y = (2) 2 = 4. If x = 0, then y = 0. If x = –1, then y = (–1) 2 = 1. If x = 1, then y = (1) 2 = 1. If x = 3, then y = (3) 2 = 9.

17. 17 Example – Solution b. We construct a table of ordered pairs for the equation x = y 2. Because y is squared, it will be more convenient to substitute values for y and compute the corresponding values for x. If x = –2, then y = (–2) 2 = 4. If x = 2, then y = (2) 2 = 4. If x = 0, then y = 0. If x = –1, then y = (–1) 2 = 1. If x = 1, then y = (1) 2 = 1. If x = 3, then y = (3) 2 = 9. cont’d

18. 18 Example – Solution From the table we can see that each input value x does not determine exactly one output value y. The relation is not a function. cont’d

19. 19 Evaluate a function written in function notation 3.

20. 20 Evaluating a Function There is a special notation for functions that uses the symbol f (x), read as “f of x.” Function Notation The notation y = f (x) denotes that the variable y is a function of x.

21. 21 Example Let y = f (x) = 2x – 3 and find a. f (3) b. f (–1) c. f (0) d. the value of x for which f (x) = 5.

22. 22 Example a) – Solution We replace x with 3. f (x) = 2x – 3 f (3) = 2(3) – 3 = 6 – 3 = 3

23. 23 Example b) – Solution We replace x with –1. f (x) = 2x – 3 f (–1) = 2(–1) – 3 = –2 – 3 = –5 cont’d

24. 24 Example c) – Solution We replace x with 0. f (x) = 2x – 3 f (0) = 2(0) – 3 = 0 – 3 = –3 cont’d

25. 25 Example d) – Solution We replace f (x) with 5 and solve for x. f (x) = 2x – 3 5 = 2x – 3 8 = 2x 4 = x Add 3 to both sides. Divide both sides by 2. cont’d

26. 26 Graph a function and determine its domain and range 4.

27. 27 Graphing a Function Not all functions are linear functions. For example, the graph of the absolute value function defined by the equation y = | x | is not a line, as the next example will show.

28. 28 Example Graph f (x) = | x | and determine the domain and range. Solution: We begin by setting up a table of values. Figure 3-42

29. 29 Example – Solution From Figure 3-42, we can determine that the domain is the set of all real numbers R. This makes sense because we can find the absolute value of any real number. From the graph, we can see that the range is the set of values where y  0. We write this as {y | y is a real number and y  0}. The absolute value of any real number is always positive or 0. cont’d

30. 30 Determine whether a graph represents a function 5.

31. 31 Does a Graph Represent a Function? A vertical line test can be used to determine whether the graph of an equation represents a function. If any vertical line intersects a graph more than once, the graph cannot represent a function, because to one number x, there would correspond more than one value of y. The graph in Figure 3-43(a) represents a function, because every vertical line that intersects the graph does so exactly once. Figure 3-43(a)

32. 32 Does a Graph Represent a Function? The graph in Figure 3-43(b) does not represent a function, because some vertical lines intersect the graph more than once. Figure 3-43(b)

33. 33 Example Determine whether each graph represents a function.

34. 34 Example – Solution a. Since any vertical line that intersects the graph does so only once, the graph represents a function. b. Since some vertical lines that intersect the graph do so twice, the graph does not represent a function. c. Since some vertical lines that intersect the graph do so twice, the graph does not represent a function. d. Since any vertical line that intersects the graph does so only once, the graph represents a function.