Lesson 7-1. Roots and Radicals Expressions

Transcription

1 Lesson 7-1 Roots and Radicals Epressions

2 Radical Sign inde Radical Sign n a Radicand

3 Eample 1 Page 66 #6 Find all the real cube roots of (0.50) is the cube root of 0.15.

4 Eample 1 Page 66 #1 10,000 Find all the real fourth roots of , , , ,

5 Eample Page 66 #18 Find each real-number root of ( 4 4 4) ( 4) 4

6 Eample Page 66 #0 4 Find each real-number root of ( ) = No Real Roots

7 Eample Page 66 # Simplify 0.5 when needed. 6. Use absolute value symbols

8 Eample Page 66 #4 Simplify 48 64b when needed.. Use absolute value symbols 48 64b 48 8 b 4 8b

9 Eample Page 66 #8 Simplify 5 y 10 when needed.. Use absolute value symbols 5 y ( ) ( y ) y 5

10 Eample 4 Page 66 #0 4 The formula for the volume of a sphere is v r. Find the radius to the nearest hundredth of a sphere with volume of 0 ft³ r 4 r 60 4 r r r r r 1.68ft

11 Lesson 7-, Part 1 Multiplying and Dividing Radical Epression

12 Multiplying Radical Epression n a n b n ab

13 Eample 1 Page 71, #4 Multiply if possible. Then simplify

14 Eample 1 Page 71, #8 Multiply if possible. Then simplify

15 Eample Page 71, #16 Simplify. Assume that all the variables are positive. y y y 4y

16 Eample Page 71, #18 Multiply and simplify. Assume that all the variables are positive. 8y 40y 5 8y 40y 5 6 y ( )(5)( y ) y 5 y 8y 5 y

17 Eample Page 71, # Multiply and simplify. Assume that all the variables are positive. y 15 5 y 7 15y y 0

18 Lesson 7-, Part Multiplying and Dividing Radical Epression

19 Dividing Radical Epression n n a b n a b

20 Eample 4 Page 71, #4 Divide and Simplify. Assume all variables are positive. 48 y 48 y 48 1 y y y 4 y

21 Eample 4 Page 71, #6 Divide and Simplify. Assume all variables are positive. 50y 7 y 50 y y y y 15y 5 y y

22 Rationalize the Denominator Rationalizing the denominator is to remove the radical in the denominator.

23 Eample 5 Page 71, #8 Rationalize the denominator of each epression. Assume that all the variables are positive

24 Eample Page 71, #0 5 Rationalize the denominator of each epression. Assume that all the variables are positive

25 Eample 5 Page 71, # Rationalize the denominator of each epression. Assume that all the variables are positive

26 Lesson 7- Binomial Radical Epression

27 Like Radicals Like Radicals are radical epressions that have the same inde and the same radicand. Like Radicals Unlike Radicals

28 Eamples 1 and Page 76, # Add or subtract if possible. 6 (6 ) 4

29 Eamples 1 and Page 76, #6 Add or subtract if possible. 7 (7 ) 5

30 Eample Page 76, #8 Simplify

31 Eample Page 76, #1 Simplify

32 Eample 4 Page 76, #14 Multiply

33 Eample 4 Page 76, #18 Multiply

34 Eample 5 Page 76, #0 Multiply each pair of conjugates

35 Eample 6 Page 76, #4 Rationalize each denominator. Simplify the answer

36 Eample 6 Page 76, #

37 Lesson 7-4, Part 1 Rational Eponents

38 Rational Eponent Radical Form n a Eponent Form 1 a n n m a m a n

39 Eample Page 8, #18 and #4 Write the epression in eponential form ( 10) 4 c c 4 1 c

40 Properties of Rational Eponents Property a m a n a m n Eample a m n a mn ab m a m b m

41 Properties of Rational Eponents Property Eample a m a 1 m a 1 m m a z y z y

42 Properties of Rational Eponents Property Eample a a m n a mn 1 1 a b m a b m m

43 Eample Page 8, #14 and #16 Write the epression in radical form. y y y

44 Eample 1 Page 8, #6 Simplify each epression

45 Eample 1 Page 8, # Simplify each epression

46 Eample 4 Page 8, #4 Simplify the number

47 Lesson 7-4, Part Rational Eponents

48 Eample 5 Page 8, #4 Write the epression in simplest form

49 Eample 5 Page 8, #46 Write the epression in simplest form. 1 y 6 y y y y 4

50 Eample 5 Page 8, #48 Write the epression in simplest form y 4 y y y y

51 Lesson 7-5 Solving Radical Equations

52 Solving Radical Equations Radical equation is an equation that has a variable in a radicand or has a variable with a rational eponent. To solve a radical equation, isolate the radical on one side of the equation and then raise both sides of the equation to the same power

53 Eample 1 Page 88, # Solve

54 Eample 1 Page 88, # 1 1 1

55 Eample 1 Page 88, #6 Solve

56 Eample 1 Page 88, #6

57 Solving equations with Rational Eponents Isolate the eponent and raise each side of the equation by its reciprocal. If the denominator of the reciprocal is even use absolute value signs.

58 Eample Page 88, #10 Solve

59 Eample Page 88, #

60 Eample Page 88, #1 Solve

61 Eample Page 88, #

62 Eample Page 88, #

63 Solving Equation with Etraneous Solutions Etraneous solutions can be introduced when you raise both sides of an equation to a power. So you need to check the solutions by substituting all solutions into the original equation.

64 Eample 4, Page 88, #18 Solve. Check for etraneous solutions

65 Eample 4, Page 88, #

66 Eample 4, Page 88, #

67 Eample 4, Page 88, #18 Check for etraneous solutions

68 Eample 5, Page 89, # Solve. Check for etraneous solutions

69 Eample 5, Page 89, #

70 Eample 5, Page 89, # Check for etraneous solutions ( 5) 5 ( ) () () () ( 10 5) 5 ( 10) ( 5) ( 5) ( 5)

71 Lesson 7-6 Functions Operations

72 Adding and Subtracting Functions f g f g f g f g

73 Eample 1 Page 94, # Let f ( ) 5 and g( ). Perform the function operation. g f g f g) ( f( ) ( 5) 5

74 Eample 1 Page 94, #8 Let f ( ) 5 and g( ). Perform the function operation. ( f g)( ) f g f( ) g ( ) 5 5

75 Multiplying and Dividing Functions f g f g f g f ( ), g ( ) 0 g( )

76 Eample Page 94, #4 Let f ( ) 5 and g( ). Perform the function operation. ( f g)( ) f g f( ) g( ) 5 5

77 Eample Page 95, #18 Let f ( ) and g( ) 1. Perform the function operation and then find the domain. g f g) ( f ( ) g ( ) f( ) 1 1 ( 1)( )

78 Eample Page 95, #18 1 ( 1)( ) 1 ( 1( ) ) 1 Domain: all real numbers, ecept and 1

79 Composite Functions When you combine two function you form a composite function. Notation g f ( ) g( f ( ) ) Evaluate the inner function f() first. The use your answer as the input of the outer function g().

80 Eample, Page 95, #4 Let g( ) and h( ) 4. Evaluate the epression. h g hg( ) h( ) 4 ( ) h g( ) 4( )

81 Eample, Page 95, #8 Let g( ) and h( ) 4. Evaluate the epression. g g g g( ) g( ) ( ) 4 gg( ) 4() 1

82 Eample, Page 95, #40 Let f ( ) and g( ). Evaluate the epression. g f c g f ( ) g( ) gf( c) c

83 Lesson 7-7 Inverse Relations and Functions

84 Inverse Interchange the -value and y-value to get the inverse.

85 Eample 1 Page 404, # Find the inverse of each relation. Graph the given relation and its inverse. y A. Graph the relation Function No repeating -values

86 Eample 1 Page 404, # y B. Find the inverse of the relation and graph it. Inverse Relation y Function No repeating -values

87 Line y Line y is the perpendicular bisector of each segment connecting a point in the relation with the corresponding point in its inverse. The graph of the inverse is a reflection in the line y of the graph of the relation.

88 Eample 1 Page 404, #4 Find the inverse of each relation. Graph the given relation and its inverse. y 1 0 A. Graph the relation Function No repeating -values

89 Eample 1 Page 404, #4 y 1 0 B. Find the inverse of the relation and graph it. Inverse Relation y 1 0 Not a Function Repeating -values

90 Eample Page 404, #6 Find the inverse of the function. Is the inverse a function? y 1 Step 1 Interchange the and y y 1

91 Eample Page 404, #6 Step Solve for y y 1 1 y Step Is the inverse a function Yes, no repeating -values original is linear 1 y 1 y 1 y

92 Eample Page 404, #1 Find the inverse of the function. Is the inverse a function? y 4 Step 1 Interchange the and y y 4

93 Eample Page 404, #1 Step Solve for y y 4 4 y y 4 y 4 4 y 4 y y 4

94 Eample Page 404, #1 Step Is the inverse a function No, repeating -values original is a quadratic

95 Eample Page 404, #14 Graph each relation and its inverse. y Inverse y y y y y 0 1 1

96 Eample Page 404, #0 Graph each relation and its inverse. y ( 1) Inverse ( y1) y1 y1 y

97 Eample 4 Page 404, #4 For each function f, find f 1 and the domain and range of f and f 1. Determine whether f 1 is a function. f ( ) 5 Domain of f y 5 5 Range of f y y 0 y Inverse y5 y 5 5 Domain of f 1 Range of f 1 0 y y 5 f 1 is a function

98 Eample 4 Page 404, #8 For each function f, find f 1 and the domain and range of f and f 1. Determine whether f 1 is a function. f y ( ) 1 Domain of f 1 {all real numbers} Range of f y y1 Inverse y 1 1 y y 1 y 1 Domain of f 1 Range of f 1 1 {all real numbers} f 1 is not a function

99 Lesson 7-8 Graphing Radical Functions

100 Eample 1 Page 411, # Graph each function. y y y

101 Eample 1 Page 411, #8 Graph each function. y 4 y y 4 0 8

102 Eample 1 Page 411, #10 Graph each function. y 0.5 y y 8.71

103 Basic Graph Functions Linear y y Graph of f() Inverse Graph of f 1 Domain: Range: Function: All Real Numbers All Real Numbers Yes All Real Numbers All Real Numbers Yes

104 Basic Graph Functions Quadratic y y Graph of f() Inverse Graph of f 1 Domain: Range: Function: All Real Numbers y y 0 Yes 0 All Real Numbers No

105 Basic Graph Functions Cubic y y Graph of f() Inverse Graph of f 1 Domain: Range: Function: All Real Numbers All Real Numbers Yes All Real Numbers All Real Numbers Yes

106 Basic Graph Functions Absolute y y 0 0 Graph of f() Inverse Graph of f 1 Domain: Range: Function: All Real Numbers y y 0 Yes 0 All Real Numbers No

107 Basic Graph Functions Square Root y y Graph of f() Inverse Graph of f 1 Domain: Range: 0 0 y y 0 y y 0 Function: Yes Yes