Lesson 7-1 Roots and Radicals Epressions
Radical Sign inde Radical Sign n a Radicand
Eample 1 Page 66 #6 Find all the real cube roots of 0.15 0.15 0.15 0.15 0.50 (0.50) 0.15 0.50 is the cube root of 0.15.
Eample 1 Page 66 #1 10,000 Find all the real fourth roots of. 81 4 10,000 81 4 10,000 81 4 4 4 10,000 81 4 10,000 4 10 81
Eample Page 66 #18 Find each real-number root of 64. 64 ( 4 4 4) ( 4) 4
Eample Page 66 #0 4 Find each real-number root of 81. 4 81 4 ( ) = No Real Roots
Eample Page 66 # Simplify 0.5 when needed. 6. Use absolute value symbols 0.5 6 6 0.50 0.50 0.50
Eample Page 66 #4 Simplify 48 64b when needed.. Use absolute value symbols 48 64b 48 8 b 4 8b
Eample Page 66 #8 Simplify 5 y 10 when needed.. Use absolute value symbols 5 y 10 5 5 ( ) ( y ) y 5
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. 0 0 1 4 r 4 r 60 4 r 60 4 4 4 r 4.775 r r 4.775 r 1.68ft
Lesson 7-, Part 1 Multiplying and Dividing Radical Epression
Multiplying Radical Epression n a n b n ab
Eample 1 Page 71, #4 Multiply if possible. Then simplify 4 4 8 4 8 4 5 8 4 4
Eample 1 Page 71, #8 Multiply if possible. Then simplify 1 18 1 18 6 1 4 18 9
Eample Page 71, #16 Simplify. Assume that all the variables are positive. y y 4 6 64 6 6 4 4 y 4y
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
Eample Page 71, # Multiply and simplify. Assume that all the variables are positive. y 15 5 y 7 15y y 0
Lesson 7-, Part Multiplying and Dividing Radical Epression
Dividing Radical Epression n n a b n a b
Eample 4 Page 71, #4 Divide and Simplify. Assume all variables are positive. 48 y 48 y 48 1 y 16 16 y y 4 y
Eample 4 Page 71, #6 Divide and Simplify. Assume all variables are positive. 50y 7 y 50 y y 7 7 50 y y 15y 5 y 5 5 5 y
Rationalize the Denominator Rationalizing the denominator is to remove the radical in the denominator.
Eample 5 Page 71, #8 Rationalize the denominator of each epression. Assume that all the variables are positive 5 8 5 8 8 8 40 8 40 8 4 10 8 10 8 10 4
Eample Page 71, #0 5 Rationalize the denominator of each epression. Assume that all the variables are positive. 5 5 45
Eample 5 Page 71, # Rationalize the denominator of each epression. Assume that all the variables are positive 5 15 60 1 5 60 1 4 5 5 5 5 5
Lesson 7- Binomial Radical Epression
Like Radicals Like Radicals are radical epressions that have the same inde and the same radicand. Like Radicals Unlike Radicals 5 5 5 5 5 5 5 5 5 7 5 5 5 7 5
Eamples 1 and Page 76, # Add or subtract if possible. 6 (6 ) 4
Eamples 1 and Page 76, #6 Add or subtract if possible. 7 (7 ) 5
Eample Page 76, #8 Simplify. 14 0 15 14 5 5 5 8 5 15 5 8 15 5 1 5 0 4 5 5 5 5 15 5 5 5 5 5 5
Eample Page 76, #1 Simplify. 4 4 48 4 4 4 4 4 48 8 6 4 4 4 4 4 5 4 4 4 8 4
Eample 4 Page 76, #14 Multiply 7 1 7 6 7 7 49 7 7 1 7 7
Eample 4 Page 76, #18 Multiply 5 4 5 6 10 6 10 9 4 0 1 10 18 8 1 10
Eample 5 Page 76, #0 Multiply each pair of conjugates. 4 4 16 4 9 16 1 4
Eample 6 Page 76, #4 Rationalize each denominator. Simplify the answer. 4 4 4
Eample 6 Page 76, #4 4 1 8 9 9 7 4
Lesson 7-4, Part 1 Rational Eponents
Rational Eponent Radical Form n a Eponent Form 1 a n n m a m a n
Eample Page 8, #18 and #4 Write the epression in eponential form. 10 1 ( 10) 4 c c 4 1 c
Properties of Rational Eponents Property a m a n a m n Eample 1 1 5 5 5 5 5 a m n a mn 1 4 1 4 4 9 ab m a m b m 1 1 1 1 4 4
Properties of Rational Eponents Property Eample a m a 1 m 16 1 1 1 1 4 16 a 1 m m a z y z y
Properties of Rational Eponents Property Eample a a m n a mn 1 1 a b m a b m m 1 1 1 5 5 5 1 8 8
Eample Page 8, #14 and #16 Write the epression in radical form. y 9 8 1 1 y y 9 8 9 8 1.5
Eample 1 Page 8, #6 Simplify each epression. 1 1 1 1 6 6 6
Eample 1 Page 8, # Simplify each epression. 1 7 1 1 1
Eample 4 Page 8, #4 Simplify the number. 4 5 1 4 5 1 1 4 4 5 5 1 16
Lesson 7-4, Part Rational Eponents
Eample 5 Page 8, #4 Write the epression in simplest form. 1 7 9 1 1 7 9 1 9
Eample 5 Page 8, #46 Write the epression in simplest form. 1 y 6 y 1 6 6 1 1 y y 4 6 1 y 4
Eample 5 Page 8, #48 Write the epression in simplest form. 1 4 1 11 4 1 y 4 y 1 4 1 y 1 4 6 4 y 9 1 9 y
Lesson 7-5 Solving Radical Equations
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
Eample 1 Page 88, # Solve. 4 1 4 4 4 4 4 4 1
Eample 1 Page 88, # 1 1 1
Eample 1 Page 88, #6 Solve. 6 0 6 6 6 4
Eample 1 Page 88, #6
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.
Eample Page 88, #10 Solve. 4 81 4 7 4 4 4 7 7 4
Eample Page 88, #10 7 4 4 4 81 78
Eample Page 88, #1 Solve. 4 11 4 8 4 8 4 8
Eample Page 88, #1 4 8 4 4 4 4
Eample Page 88, #1 4 4 0 0
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.
Eample 4, Page 88, #18 Solve. Check for etraneous solutions. 7 5 7 5 7 5 7 10 5
Eample 4, Page 88, #18 7 10 5 0 11 18 11 18 0
Eample 4, Page 88, #18 11 18 0 9 0 0 9 0 9
Eample 4, Page 88, #18 Check for etraneous solutions. 9 7 5 7 5 7 5 9 5 5 8 9 7 5 9 16 5 9 4 5 9 9 9
Eample 5, Page 89, # Solve. Check for etraneous solutions. 1 1 4 5 5 0 1 1 4 5 5 4 4 1 1 1 1 4 5 5
Eample 5, Page 89, # 4 4 1 1 1 1 4 5 5 5 5 10 5 5 1 0 0 10 0 0 10 0 10
Eample 5, Page 89, # Check for etraneous solutions. 10 1 1 4 5 5 0 1 1 4 ( 5) 5 ( ) 0 1 1 4 () 5 4 0 1 1 4 () 9 0 1 1 4 () 0 1 1 0 1 1 4 5 5 0 1 1 4 ( 10 5) 5 ( 10) 0 1 1 4 ( 5) 5 0 0 1 1 4 ( 5) 5 0 1 1 4 ( 5) 5 0 1 1 5 5 0
Lesson 7-6 Functions Operations
Adding and Subtracting Functions f g f g f g f g
Eample 1 Page 94, # Let f ( ) 5 and g( ). Perform the function operation. g f g f g) ( f( ) ( 5) 5
Eample 1 Page 94, #8 Let f ( ) 5 and g( ). Perform the function operation. ( f g)( ) f g f( ) g ( ) 5 5
Multiplying and Dividing Functions f g f g f g f ( ), g ( ) 0 g( )
Eample Page 94, #4 Let f ( ) 5 and g( ). Perform the function operation. ( f g)( ) f g f( ) g( ) 5 5
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)( )
Eample Page 95, #18 1 ( 1)( ) 1 ( 1( ) ) 1 Domain: all real numbers, ecept 0 10 1 and 1
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().
Eample, Page 95, #4 Let g( ) and h( ) 4. Evaluate the epression. h g hg( ) h( ) 4 ( ) 4 4 4 h g( ) 4( ) 4 16 4 0
Eample, Page 95, #8 Let g( ) and h( ) 4. Evaluate the epression. g g g g( ) g( ) ( ) 4 gg( ) 4() 1
Eample, Page 95, #40 Let f ( ) and g( ). Evaluate the epression. g f c g f ( ) g( ) gf( c) c
Lesson 7-7 Inverse Relations and Functions
Inverse Interchange the -value and y-value to get the inverse.
Eample 1 Page 404, # Find the inverse of each relation. Graph the given relation and its inverse. y 1 4 0 1 A. Graph the relation Function No repeating -values
Eample 1 Page 404, # y 1 4 0 1 B. Find the inverse of the relation and graph it. Inverse Relation y 0 1 1 4 Function No repeating -values
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.
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
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
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
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
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
Eample Page 404, #1 Step Solve for y y 4 4 y y 4 y 4 4 y 4 y y 4
Eample Page 404, #1 Step Is the inverse a function No, repeating -values original is a quadratic
Eample Page 404, #14 Graph each relation and its inverse. y Inverse y y y y y 0 1 1
Eample Page 404, #0 Graph each relation and its inverse. y ( 1) Inverse ( y1) y1 y1 y 4 1 0 1 1 4
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
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
Lesson 7-8 Graphing Radical Functions
Eample 1 Page 411, # Graph each function. y y y 0 1 1 4 0
Eample 1 Page 411, #8 Graph each function. y 4 y y 4 0 8
Eample 1 Page 411, #10 Graph each function. y 0.5 y 4.50 0 0 y 8.71
Basic Graph Functions Linear y y 0 0 1 1 1 1 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
Basic Graph Functions Quadratic y y 0 0 4 4 Graph of f() Inverse Graph of f 1 Domain: Range: Function: All Real Numbers y y 0 Yes 0 All Real Numbers No
Basic Graph Functions Cubic y y 0 0 1 1 1 1 8 8 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
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
Basic Graph Functions Square Root y y 0 0 4 9 Graph of f() Inverse Graph of f 1 Domain: Range: 0 0 y y 0 y y 0 Function: Yes Yes