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COLLEGE ALGEBRA

Learn about relations and functions, determining functions from graphs or equations, function notation, and motion problems in college algebra.

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COLLEGE ALGEBRA

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  1. COLLEGE ALGEBRA LIAL HORNSBY SCHNEIDER

  2. 2.3 Functions Relations and Functions Domain and Range Determining Functions from Graphs or Equations Function Notation Increasing, Decreasing, and Constant Functions

  3. Relation A relation is a set of ordered pairs.

  4. Function A function is a relation in which, for each distinct value of the first component of the ordered pair, there is exactly one value of the second component.

  5. Motion Problems Note The relation from the beginning of this section representing the number of gallons of gasoline and the corresponding cost is a function since each x-value is paired with exactly one y-value. You would not be happy if you and a friend each pumped 10 gal of regular gasoline at the same station and your bills were different.

  6. DECIDING WHETHER RELATIONS DEFINE FUNCTIONS Example 1 Decide whether the relation defines a function. Solution Relation F is a function, because for each different x-value there is exactly one y-value. We can show this correspondence as follows. x-values of F y-values of F

  7. DECIDING WHETHER RELATIONS DEFINE FUNCTIONS Example 1 Decide whether the relation defines a function. Solution As the correspondence shows below, relation G is not a function because one first component corresponds to more than one second component. x-values of G y-values of G

  8. DECIDING WHETHER RELATIONS DEFINE FUNCTIONS Example 1 Decide whether the relation defines a function. Solution In relation H the last two ordered pairs have the same x-value paired with two different y-values, so H is a relation but not a function. Different y-values Not a function Same x-values

  9. Mapping Relations and functions can also be expressed as a correspondence or mapping from one set to another. In the example below the arrows from 1 to 2 indicates that the ordered pair (1, 2) belongs to F. Each first component is paired with exactly one second component. x-axis values y-axis values 1 –2 3 2 4 –1

  10. Mapping In the mapping for relations H, which is not a function, the first component –2 is paired with two different second components, 1 and 0. x-axis values y-axis values –4 –2 1 0

  11. Relations Note Another way to think of a function relationship is to think of the independent variable as an input and the dependent variable as an output.

  12. Domain and Range In a relation, the set of all values of the independent variable (x) is the domain. The set of all values of the dependent variable (y) is the range.

  13. FINDING DOMAINS AND RANGES OF RELATIONS Example 2 Give the domain and range of the relation. Tell whether the relation defines a function. a. The domain, the set of x-values, is {3, 4, 6}; the range, the set of y-values is {–1, 2, 5, 8}. This relation is not a function because the same x-value, 4, is paired with two different y-values, 2 and 5.

  14. FINDING DOMAINS AND RANGES OF RELATIONS Example 2 Give the domain and range of the relation. Tell whether the relation defines a function. b. 4 6 7 –3 100 200 300 The domain is {4, 6, 7, –3}; the range is {100, 200, 300}. This mapping defines a function. Each x-value corresponds to exactly one y-value.

  15. FINDING DOMAINS AND RANGES OF RELATIONS Example 2 Give the domain and range of the relation. Tell whether the relation defines a function. c. This relation is a set of ordered pairs, so the domain is the set of x-values {–5, 0, 5} and the range is the set of y-values {2}. The table defines a function because each different x-value corresponds to exactly one y-value.

  16. FINDING DOMAINS AND RANGES FROM GRAPHS Example 3 Give the domain and range of each relation. y a. The domain is the set of x-values which are {–1, 0, 1, 4}. The range is the set of y-values which are {–3, –1, 1, 2}. (1, 2) (–1, 1) x (0, –1) (4, –3)

  17. FINDING DOMAINS AND RANGES FROM GRAPHS Example 3 Give the domain and range of each relation. y The x-values of the points on the graph include all numbers between –4 and 4, inclusive. The y-values include all numbers between –6 and 6, inclusive. b. 6 x –4 4 The domain is [–4, 4]. The range is [–6,6]. –6

  18. FINDING DOMAINS AND RANGES FROM GRAPHS Example 3 Give the domain and range of each relation. y c. The arrowheads indicate that the line extends indefinitely left and right, as well as up and down. Therefore, both the domain and the range include all real numbers, written (–, ). x

  19. FINDING DOMAINS AND RANGES FROM GRAPHS Example 3 Give the domain and range of each relation. y d. The arrowheads indicate that the line extends indefinitely left and right, as well as upward. The domain is (–, ). Because there is at least y-value, –3, the range includes all numbers greater than, or equal to –3 or [–3, ). x

  20. Agreement on Domain Unless specified otherwise, the domain of a relation is assumed to be all real numbers that produce real numbers when substituted for the independent variable.

  21. Vertical Line Test If each vertical line intersects a graph in at most one point, then the graph is that of a function.

  22. USING THE VERTICAL LINE TEST Example 4 Use the vertical line test to determine whether each relation graphed is a function. y a. (1, 2) This graph represents a function. (–1, 1) x (0, –1) (4, –3)

  23. USING THE VERTICAL LINE TEST Example 4 Use the vertical line test to determine whether each relation graphed is a function. y b. 6 This graph fails the vertical line test, since the same x-value corresponds to two different y-values; therefore, it is not the graph of a function. x –4 4 –6

  24. USING THE VERTICAL LINE TEST Example 4 Use the vertical line test to determine whether each relation graphed is a function. y c. This graph represents a function. x

  25. USING THE VERTICAL LINE TEST Example 4 Use the vertical line test to determine whether each relation graphed is a function. y d. This graph represents a function. x

  26. Relations Note Graphs that do not represent functions are still relations. Remember that all equations and graphs represent relations and that all relations have a domain and range.

  27. IDENTIFYING FUNCTIONS, DOMAINS, AND RANGES Example 5 Decide whether each relation defines a function and give the domain and range. a. Solution Since y is always found by adding 4 to x, each value of x corresponds to just one value of y and the relation defines a function. x can be any real number, so the domain is or Since y is always 4 more than x, y may also be any real number, and so the range is

  28. IDENTIFYING FUNCTIONS, DOMAINS, AND RANGES Example 5 Decide whether each relation defines a function and give the domain and range. b. Solution For any choice of x in the domain, there is exactly one corresponding value for y (the radical is a nonnegative number), so this equation is a function. Since the equation involves a square root, the quantity under the radical cannot be negative.

  29. IDENTIFYING FUNCTIONS, DOMAINS, AND RANGES Example 5 Decide whether each relation defines a function and give the domain and range. b. Solution Solve the inequality. Add 1. Divide by 2. Domain is

  30. IDENTIFYING FUNCTIONS, DOMAINS, AND RANGES Example 5 Decide whether each relation defines a function and give the domain and range. b. Solution Solve the inequality. Add 1. Divide by 2. Because the radical is a non-negative number, as x takes values greater than or equal to ½ , the range is y ≥ 0 or

  31. IDENTIFYING FUNCTIONS, DOMAINS, AND RANGES Example 5 Decide whether each relation defines a function and give the domain and range. c. Solution Ordered pairs (16, 4) and (16, –4) both satisfy the equation. Since one value of x, 16, corresponds to two values of y, this equation does not define a function. The domain is Any real number can be squared, so the range of the relation is

  32. IDENTIFYING FUNCTIONS, DOMAINS, AND RANGES Example 5 Decide whether each relation defines a function and give the domain and range. d. Solution The ordered pairs (1, 0), (1, –1), (1, –2), and (1, –3) all satisfy the inequality. An inequality rarely defines a function. Since any number can be used for x or for y, the domain and range are the set of real numbers or

  33. IDENTIFYING FUNCTIONS, DOMAINS, AND RANGES Example 5 Decide whether each relation defines a function and give the domain and range. e. Solution Substituting any value in for x, subtracting 1 and then dividing it into 5, produces exactly one value of y for each value in the domain. This equation defines a function.

  34. IDENTIFYING FUNCTIONS, DOMAINS, AND RANGES Example 5 Decide whether each relation defines a function and give the domain and range. e. Solution Domain includes all real numbers except those making the denominator 0. Add 1. The domain includes all real numbers except 1 and is written The range is the interval

  35. Function Notation When a function  is defined with a rule or an equation using x and y for the independent and dependent variables, we say “y is a function of x” to emphasize that ydepends on x. We use the notation. called a function notation, to express this and read (x) as “ of x.” The letter is he name given to this function. For example, if y = 9x– 5, we can name the function  and write

  36. Function Notation Note that (x) is just another name for the dependent variable y. Fore example, if y = (x) = 9x – 5 and x = 2, then we find y, or (2), by replacing x with 2. The statement “if x = 2, the y = 13” represents the ordered pair (2, 13) and is abbreviated with the function notation as

  37. Function Notation Read “ of 2” or “ at 2.” Also, and These ideas can be illustrated as follows. Name of the function Defining expression Value of the function Name of the independent variable

  38. Variations of the Definition of Function • A function is a relation in which, for each distinct value of the first component of the ordered pairs, there is exactly one value of the second component. • A function is a set of ordered pairs in which no first component is repeated. • A function is a rule or correspondence that assigns exactly one range value to each distinct domain value.

  39. CautionThe symbol (x) does not indicate “ times x,” but represents the y-value for the indicated x-value. As just shown, (2) is the y-value that corresponds to the x-value 2.

  40. USING FUNCTION NOTATION Example 6 Let (x) = –x2 + 5x –3 and g(x) = 2x + 3. Find and simplify. a. Solution Replace x with 2. Apply the exponent; multiply. Add and subtract. Thus, (2) = 3; the ordered pair (2, 3) belongs to .

  41. USING FUNCTION NOTATION Example 6 Let (x) = –x2 + 5x –3 and g(x) = 2x + 3. Find and simplify. b. Solution Replace x with q.

  42. USING FUNCTION NOTATION Example 6 Let (x) = x2 + 5x –3 and g(x) = 2x + 3. Find and simplify. c. Solution Replace x with a + 1.

  43. USING FUNCTION NOTATION Example 7 For each function, find (3). a. Solution Replace x with 3.

  44. USING FUNCTION NOTATION Example 7 For each function, find (3). b. Solution For  = {(–3, 5), (0, 3), (3, 1), (6,–9)}, we want (3), the y-value of the ordered pair where x = 3. As indicated by the ordered pair (3, 1), when x = 3, y = 1,so(3) = 1.

  45. USING FUNCTION NOTATION Example 7 For each function, find (3). c. Domain Range –2 3 10 6 5 2 Solution In the mapping, the domain element 3 is paired with 5 in the range, so (3) = 5.

  46. USING FUNCION NOTATION Example 7 For each function, find (3). d. 4 Solution Start at 3 on the x-axis and move up to the graph. Then, moving horizontally to the y-axis gives 4 for the corresponding y-value. Thus (3) = 4. 2 0 4 3 2

  47. Finding an Expression for (x) • Consider an equation involving x • and y. Assume that y can be expressed as a function  of x. • To find an expression for (x): • Solve the equation for y. • Replace y with (x).

  48. WRITING EQUATIONS USING FUNCTION NOTATION Example 8 Assume that y is a function of x. Rewrite the function using notation. a. Solution Let y = (x) Now find (–2) and (a). Let x = –2

  49. WRITING EQUATIONS USING FUNCTION NOTATION Example 8 Assume that y is a function of x. Rewrite the function using notation. a. Solution Let y = (x) Now find (–2) and (a). Let x = a

  50. WRITING EQUATIONS USING FUNCTION NOTATION Example 8 Assume that y is a function of x. Rewrite the function using notation. b. Solution Solve for y. Multiply by–1; divide by 4.

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