Chapter 4: Introduction to Relations and Functions

1. Chapter 4: Introduction to Relations and Functions

2. 4.1 Introduction to Relations

3. Definition of a Relation in x and y Any set of ordered pairs (x,y) is called a relation in x and y. • The set of first components in the ordered pairs is called the domain of the relation. • The set of second components in the ordered pairs is called the range of the relation.

4. Defining a Relation • Ordered pairs: { (1,2),(–3,4),(1,–4),(3,4)} • A graph: • A correspondence: • An equation:The solutions to the equation define the set of ordered pairs, and the solutions can be graphed

5. Exercise 1 List the domain and range. A) Domain {6, –2, 8, –5, –2, 8}; range {4, 2, 9, 1, 6, 1}  B) Domain {6, –2, 8, –5}; range {4, 2, 9, 1, 6}  C) Domain {6, –2, 8, –5); range {4, 2, 9, 1, 6, 1}  D) None of the above

6. Exercise 2 The following table describes the temperature (y) inside an oven, in degrees, x minutes after it was turned on. What is the domain of this relation?   A) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}  B) (72, 133, 186, 237, 285, 333, 379, 425}  C) {x | 0 <x< 9}  D) {y | 72 <y< 425}

7. Exercise 3 What is the range of the relation? A) {x | –20 <x< 15} B) {–20, –10, 0, 10, 15} C) {y | –40 <y< 20} D) {0, –20, –40, 20}

8. 4.2 Introduction to Functions

9. Definition of a Function Given a relation in x and y, we say “y is a function of x” if for every element x in the domain there corresponds exactly one element y in the range. Functions Nonfunctions {(1,3),(2,5),(6,3)} {(1,3),(2,5),(1,4)}

10. The Vertical Line Test • Consider a relation defined by a set of points (x,y) in a rectangular coordinate system. The graph defines y as a function of x if no vertical line intersects the graph in more than one point. • If any vertical line drawn through the graph of a relation intersects the graph in more than one point, then the relation does not define y as a function of x.

11. Function Notation Functions are often defined by equations. The equation y = 5x can be written in function notation as f(x) = 5x, where f is the name of the function, x is the input value from the domain of the function, and f(x) is the function value corresponding to x.

12. Evaluating a Function • A function may be evaluated at different values of x by substituting x-values from the domain into the function. • A function may be evaluated at algebraic expressions.

13. We can find function values by looking at a graph of a function. The value of f(a) refers to the y-coordinate of a point with x-coordinate a. h(3) = 1 h(–3) = 0 h(x) = –1 for x = 5

14. Domain of a Function Consider a function defined by y= f(x). The domain of f is the set of all x-values that when substituted into the function produce a real number. The range is the set of all y-values corresponding to the values of x in the domain. To find the domain of f, • Exclude values of x that make the denominator of a fraction zero. • Exclude values of x that make the expression within a square root negative.

15. Exercise 4 Which relation is NOT a function?   A) {(–5, 2), (3, 3), (0, 3), (11, 3)} B) {(5, 5), (3, 3), (11, 11), (0, 0)} C) {(5, 3), (3, 11), (–5, 11), (11, 3)}  D) {(5, 3), (0, 5), (3, 5), (5, 11)}

16. Exercise 5 If z(t) = 2t2 + 7t – 4, find z(–5) and z(6).  A) z(–5) = –89; z(6) = 110 B) z(–5) = 11; z(6) = 110 C) z(–5) = 53; z(6) = 75  D) z(–5) = 61; z(6) = 182

17. Exercise 6 Find the domain of the function. Write your answer in interval notation. A) B) C) (–19,16) D)

18. 4.3 Graphs of Basic Functions

19. Definition of a Linear Function and a Constant Function Let m and b represent real numbers such that m≠ 0. A function that can be written in the form f(x) = mx + b is a linear function. A function that can be written in the form f(x) = b is a constant function. Note: The graphs of linear and constant functions are lines.

20. To determine the shape of a basic function, we can plot several points to establish the pattern of the graph. Analyzing the equation itself may also provide insight to the domain, range, and shape of the function.

21. Summary of Basic Functions and Their Graphs f(x) = x Domain (–∞, ∞) Range (–∞, ∞) f(x) = x2 Domain (–∞, ∞) Range [0, ∞)

22. Summary of Basic Functions and Their Graphs f(x) = x3 Domain (–∞, ∞) Range (–∞, ∞) f(x) = x Domain (–∞, ∞) Range [0, ∞)

23. Summary of Basic Functions and Their Graphs Domain [0, ∞) Range [0, ∞) Domain Range

24. Quadratic Functions • A quadratic function can be written in the form f(x) = ax2 + bx + c, where a, b, and c are real numbers and a≠ 0. • The graph of a quadratic is in the shape of a parabola. • The leading coefficient a determines the direction of the parabola. • If a > 0, the parabola opens upward and the vertex is the minimum point. • If a < 0, the parabola opens downward and the vertex is the maximum point.

25. Finding the x- and y-Intercepts of a Function Given a function defined by y = f(x), • The x-intercepts are the real solutions to the equation f(x) = 0. • The y-intercept is given by f(0). The x-intercepts are (1,0) and (5,0). The y-intercept is (0,1).

26. Exercise 7 Which of the following is a linear function? A) B) C) D)

27. Exercise 8 Which is the graph of f(x) = x3? A) B) C) D)

28. Exercise 9 Find the x- and y-intercepts of the function. g(x) = –8x – 6 A) (0,0) B) (0, –6) and C) (–6, 0) and D) (0, –6) and (14, 0)

29. 4.4 Variation

30. Direct and Inverse Variation Let k be a nonzero constant real number. y varies directlyas x. y is directly proportional to x. y varies inverselyas x. y is inversely proportional to x. k is called the constant of variation.

31. For positive values of k, • When two variables are directly related, as one variable increases the other variable will also increase. • When two variables are inversely related, as one variable increases the other will decrease.

32. Joint Variation Let k be a nonzero constant real number. y varies jointlyas w and z. y is jointly proportional to w and z.

33. Steps to Find a Variation Model • Write a general variation model that relates the variables given in the problem. Let k represent the constant of variation. • Solve for k by substituting known values of the variables into the model from step 1. • Substitute the value of k into the original variation model from step 1.

34. Example C varies directly as the square root of d and inversely as t. If C = 12 when d is 9 and t is 6, find C if d is 16 and t is 12. Step 1: Write a general model. Step 2: Solve for k using known values.

35. Example (continued) Step 3: Substitute the value of k and the other known variables to find C.

36. Exercise 10 In the equation y = –9x, x and y vary A) Directly B) Inversely C) Jointly

37. Exercise 11 Translate the statement into symbols: The number of hours it takes to paint a house is inversely proportional to the number of people painting. Let t = number of hours, n = number of workers. A)   B) t = kn C) tnk = 1  D)

38. Exercise 12 The cost, in dollars, of filling your gas tank is directly proportional to the price per gallon of gas. If 15 gallons of gas costs $19.50, how much would 18 gallons cost? A) $22.50 B) $23.40 C) $24.50 D) $11.54

39. B A C D B A D C B A D B Exercise Answers