.1..... Rational Eponents and Radical Functions nth Roots and Rational Eponents Properties of Rational Eponents and Radicals Graphing Radical Functions Solving Radical Equations and Inequalities Performing Function Operations Inverse of a Function SEE the Big Idea Hull Speed (p. 8) White Rhino (p. 8) Concert (p. ) Mars Rover (p. 10) Constellations (p. 0) Int_Math_PE_0OP.indd 190 1/0/1 :1 PM
Maintaining Mathematical Proficienc Properties of Integer Eponents Eample 1 Simplif the epression. = + = 7 Eample Simplif the epression ( s t ). ( s t ) Product of Powers Propert Add eponents. = 7 Quotient of Powers Propert = Subtract eponents. = (s ) t = (s ) t = s t Simplif the epression. Power of a Quotient Propert Power of a Product Propert Power of a Power Propert 1.. n n... ( w z ). ( m7 m z m ) Rewriting Literal Equations Eample Solve the literal equation = 10 for. = 10 + = 10 + = 10 + = 10 + Solve the literal equation for. Write the equation. Add to each side. Simplif. = Simplif. Divide each side b. 7. + = 8. 1 = 1 9. 9 = 1 10. + = 10 11. 8 = 1. + 7 = 1 1. ABSTRACT REASONING Is the order in which ou appl properties of eponents important? Eplain our reasoning. Dnamic Solutions available at BigIdeasMath.com 191
Mathematical Practices Using Technolog to Evaluate Roots Core Concept Evaluating Roots with a Calculator Eample Square root: = 8 Cube root: = Fourth root: = Fifth root: = Mathematicall profi cient students epress numerical answers precisel. square root cube root fourth root fifth root () () () () 8 Approimating Roots Evaluate each root using a calculator. Round our answer to two decimal places. a. 0 b. 0 c. 0 d. 0 a. 0 7.07 Round down. b. c. d. 0.8 0. 0.19 Round down. Round up. Round up. (0) (0) (0) (0) 7.0710781.80199.91798.18718 Monitoring Progress 1. Use the Pthagorean Theorem to find the eact lengths of a, b, c, and d in the figure.. Use a calculator to approimate each length to the nearest tenth of an inch in Monitoring Progress Question 1.. Use a ruler to check the reasonableness of our answers in Monitoring Progress Question. 1 in. 1 in. a b 1 in. c d 1 in. 1 in. 19 Chapter Rational Eponents and Radical Functions
.1 nth Roots and Rational Eponents Essential Question How can ou use a rational eponent to represent a power involving a radical? Previousl, ou learned that the nth root of a can be represented as n a = a 1/n Definition of rational eponent for an real number a and integer n greater than 1. Eploring the Definition of a Rational Eponent CONSTRUCTING VIABLE ARGUMENTS To be proficient in math, ou need to understand and use stated definitions and previousl established results. Work with a partner. Use a calculator to show that each statement is true. a. 9 = 9 1/ b. = 1/ c. 8 = 8 1/ d. = 1/ e. 1 = 1 1/ f. Writing Epressions in Rational Eponent Form 1 = 1 1/ Work with a partner. Use the definition of a rational eponent and the properties of eponents to write each epression as a base with a single rational eponent. Then use a calculator to evaluate each epression. Round our answer to two decimal places. Sample ( ) = ( 1/ ) = /. ^(/).1981 a. ( ) b. ( ) c. ( 9 ) d. ( 10 ) e. ( 1 ) f. ( 7 ) Writing Epressions in Radical Form Work with a partner. Use the properties of eponents and the definition of a rational eponent to write each epression as a radical raised to an eponent. Then use a calculator to evaluate each epression. Round our answer to two decimal places. Sample / = ( 1/ ) = ( ).9 a. 8 / b. / c. 1 / d. 10 / e. 1 / f. 0 / Communicate Your Answer. How can ou use a rational eponent to represent a power involving a radical?. Evaluate each epression without using a calculator. Eplain our reasoning. a. / b. / c. / d. 9 / e. 1 / f. 100 / Section.1 nth Roots and Rational Eponents 19
.1 Lesson What You Will Learn Core Vocabular nth root of a, p. 19 inde of a radical, p. 19 Previous square root cube root eponent Find nth roots of numbers. Evaluate epressions with rational eponents. Solve equations using nth roots. nth Roots You can etend the concept of a square root to other tpes of roots. For eample, is a cube root of 8 because = 8. In general, for an integer n greater than 1, if b n = a, then b is an nth root of a. An nth root of a is written as n a, where n is the inde of the radical. You can also write an nth root of a as a power of a. If ou assume the Power of a Power Propert applies to rational eponents, then the following is true. (a 1/ ) = a (1/) = a 1 = a (a 1/ ) = a (1/) = a 1 = a (a 1/ ) = a (1/) = a 1 = a Because a 1/ is a number whose square is a, ou can write a = a 1/. Similarl, a = a 1/ and a = a 1/. In general, n a = a 1/n for an integer n greater than 1. UNDERSTANDING MATHEMATICAL TERMS When n is even and a > 0, there are two real roots. The positive root is called the principal root. Core Concept Real nth Roots of a Let n be an integer (n > 1) and let a be a real number. n is an even integer. n is an odd integer. a < 0 No real nth roots a < 0 One real nth root: n a = a 1/n a = 0 One real nth root: n 0 = 0 a = 0 One real nth root: n 0 = 0 a > 0 Two real nth roots: ± n a = ±a 1/n a > 0 One real nth root: n a = a 1/n Finding nth Roots Find the indicated real nth root(s) of a. a. n =, a = 1 b. n =, a = 81 a. Because n = is odd and a = 1 < 0, 1 has one real cube root. Because ( ) = 1, ou can write 1 = or ( 1) 1/ =. b. Because n = is even and a = 81 > 0, 81 has two real fourth roots. Because = 81 and ( ) = 81, ou can write ± 81 = ± or ±81 1/ = ±. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Find the indicated real nth root(s) of a. 1. n =, a = 1. n =, a = 9. n =, a = 1. n =, a = 19 Chapter Rational Eponents and Radical Functions
Rational Eponents A rational eponent does not have to be of the form 1/n. Other rational numbers, such as / and 1/, can also be used as eponents. Two properties of rational eponents are shown below. Core Concept Rational Eponents Let a 1/n be an nth root of a, and let m be a positive integer. a m/n = (a 1/n ) m = ( n a ) m a m/n 1 = a m/n = 1 (a 1/n ) m = 1 ( n a ) m, a 0 Evaluating Epressions with Rational Eponents Evaluate each epression. a. 1 / b. / COMMON ERROR Be sure to use parentheses to enclose a rational eponent: 9^(1/) 1.. Without them, the calculator evaluates a power and then divides: 9^1/ = 1.8. Rational Eponent Form a. 1 / = (1 1/ ) = = b. / 1 = / = 1 ( 1/ ) = 1 = 1 8 Radical Form 1 / = ( 1 ) = = / 1 = / = 1 ( ) = 1 = 1 8 When using a calculator to approimate an nth root, ou ma want to rewrite the nth root in rational eponent form. Approimating Epressions with Rational Eponents Evaluate each epression using a calculator. Round our answer to two decimal places. a. 9 1/ b. 1 /8 c. ( 7 ) a. 9 1/ 1. b. 1 /8. c. Before evaluating ( 7 ), rewrite the epression in rational eponent form. 9^(1/) 1.187 1^(/8).91791 7^(/).017071 ( 7 ) = 7 /.0 Monitoring Progress Evaluate the epression without using a calculator. Help in English and Spanish at BigIdeasMath.com. /. 9 1/ 7. 81 / 8. 1 7/8 Evaluate the epression using a calculator. Round our answer to two decimal places when appropriate. 9. / 10. / 11. ( 1 ) 1. ( 0 ) Section.1 nth Roots and Rational Eponents 19
Solving Equations Using nth Roots To solve an equation of the form u n = d, where u is an algebraic epression, take the nth root of each side. Solving Equations Using nth Roots Find the real solution(s) of (a) = 18 and (b) ( ) = 1. COMMON ERROR When n is even and a > 0, be sure to consider both the positive and negative nth roots of a. a. = 18 Write original equation. = Divide each side b. = Take fifth root of each side. = Simplif. The solution is =. b. ( ) = 1 Write original equation. = ± 1 Take fourth root of each side. = ± 1 Add to each side. = + 1 or = 1 Write solutions separatel..1 or 0.8 Use a calculator. The solutions are.1 and 0.8. Real-Life Application A hospital purchases an ultrasound machine for $0,000. The hospital epects the useful life of the machine to be 10 ears, at which time its value will have depreciated to $8000. The hospital uses the declining balances method for depreciation, so the annual depreciation rate r (in decimal form) is given b the formula r = 1 ( S C) 1/n. In the formula, n is the useful life of the item (in ears), S is the salvage value (in dollars), and C is the original cost (in dollars). What annual depreciation rate did the hospital use? The useful life is 10 ears, so n = 10. The machine depreciates to $8000, so S = 8000. The original cost is $0,000, so C = 0,000. So, the annual depreciation rate is r = 1 ( S C) 1/n = 1 ( 0,000) 8000 1/10 = 1 ( ) 1/10 0.17. The annual depreciation rate is about 0.17, or 1.7%. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Find the real solution(s) of the equation. Round our answer to two decimal places when appropriate. 1. 8 = 1. 1 = 1 1. ( + ) = 1 1. ( ) = 1 17. WHAT IF? In Eample, what is the annual depreciation rate when the salvage value is $000? 19 Chapter Rational Eponents and Radical Functions
.1 Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check 1. VOCABULARY Rewrite the epression a s/t in radical form. Then state the inde of the radical.. COMPLETE THE SENTENCE For an integer n greater than 1, if b n = a, then b is a(n) of a.. WRITING Eplain how to use the sign of a to determine the number of real fourth roots of a and the number of real fifth roots of a.. WHICH ONE DOESN T BELONG? Which epression does not belong with the other three? Eplain our reasoning. (a 1/n ) m ( n a ) m ( m a ) n a m/n Monitoring Progress and Modeling with Mathematics In Eercises 10, find the indicated real nth root(s) of a. (See Eample 1.). n =, a = 8. n =, a = 1 7. n =, a = 0 8. n =, a = 9. n =, a = 10. n =, a = 79 In Eercises 11 18, evaluate the epression without using a calculator. (See Eample.) 11. 1/ 1. 8 1/ 1. / 1. 81 / 1. ( ) 1/ 1. ( ) / 17. 8 / 18. 1 7/ ERROR ANALYSIS In Eercises 19 and 0, describe and correct the error in evaluating the epression. 19. 7 / = (7 1/ ) = 9 = 81 USING STRUCTURE In Eercises 1, match the equivalent epressions. Eplain our reasoning. 1. ( ) A. 1/. ( ) B. /. 1 C. 1/. D. / In Eercises, evaluate the epression using a calculator. Round our answer to two decimal places when appropriate. (See Eample.).,78. 7 19 7. 1/ 8. 8 1/ 9. 0,7 / 0. 8 / 1. ( 187 ). ( 8 ) 8 MATHEMATICAL CONNECTIONS In Eercises and, find the radius of the figure with the given volume.. V = 1 ft. V = 1 cm 0. / = ( ) = r r 9 cm = Section.1 nth Roots and Rational Eponents 197
In Eercises, find the real solution(s) of the equation. Round our answer to two decimal places when appropriate. (See Eample.). = 1. = 1080 7. ( + 10) = 70 8. ( ) = 9. = 8 0. 7 = 1. + = 100. + 0 =. 1 = 7. 1 =. MODELING WITH MATHEMATICS When the average price of an item increases from p 1 to p over a period of n ears, the annual rate of inflation r (in decimal form) is given b r = ( p 1/n p 1 ) 1. Find the rate of inflation for each item in the table. (See Eample.) Item Price in 191 Price in 01 Potatoes (lb) $0.01 $0.7 Ham (lb) $0.1 $.9 Eggs (dozen) $0.7 $1.9. HOW DO YOU SEE IT? The graph of = n is shown in red. What can ou conclude about the value of n? Determine the number of real nth roots of a. Eplain our reasoning. = a 7. NUMBER SENSE Between which two consecutive integers does 1 lie? Eplain our reasoning. 8. THOUGHT PROVOKING In 119, Johannes Kepler published his third law, which can be given b d = t, where d is the mean distance (in astronomical units) of a planet from the Sun and t is the time (in ears) it takes the planet to orbit the Sun. It takes Mars 1.88 ears to orbit the Sun. Graph a possible location of Mars. Justif our answer. (The diagram shows the Sun at the origin of the -plane and a possible location of Earth.) (1, 0) Not drawn to scale 9. PROBLEM SOLVING A weir is a dam that is built across a river to regulate the flow of water. The flow rate Q (in cubic feet per second) can be calculated using the formula Q =.7 h /, where is the length (in feet) of the bottom of the spillwa and h is the depth (in feet) of the water on the spillwa. Determine the flow rate of a weir with a spillwa that is 0 feet long and has a water depth of feet. spillwa h 0. REPEATED REASONING The mass of the particles that a river can transport is proportional to the sith power of the speed of the river. A certain river normall flows at a speed of 1 meter per second. What must its speed be in order to transport particles that are twice as massive as usual? 10 times as massive? 100 times as massive? Maintaining Mathematical Proficienc Simplif the epression. Write our answer using onl positive eponents. (Skills Review Handbook) 1.. 7. (z ). ( ) Write the number in standard form. (Skills Review Handbook). 10. 10 7. 8. 10 1 8..9 10 Reviewing what ou learned in previous grades and lessons 198 Chapter Rational Eponents and Radical Functions
. Properties of Rational Eponents and Radicals Essential Question How can ou use properties of eponents to simplif products and quotients of radicals? Reviewing Properties of Eponents Work with a partner. Let a and b be real numbers. Use the properties of eponents to complete each statement. Then match each completed statement with the propert it illustrates. Statement Propert a. a =, a 0 A. Product of Powers b. (ab) = B. Power of a Power c. (a ) = C. Power of a Product d. a a = D. Negative Eponent e. ( a b ) =, b 0 E. Zero Eponent f. a =, a 0 F. Quotient of Powers a USING TOOLS STRATEGICALLY To be proficient in math, ou need to consider the tools available to help ou check our answers. For instance, the following calculator screen shows that and 8 are equivalent. ( ())( ()) (8) g. a 0 =, a 0 G. Power of a Quotient Simplifing Epressions with Rational Eponents Work with a partner. Show that ou can appl the properties of integer eponents to rational eponents b simplifing each epression. Use a calculator to check our answers. a. / / b. 1/ / c. ( / ) d. (10 1/ ) e. 8/ 8 1/ f. 7/ 7 / Simplifing Products and Quotients of Radicals Work with a partner. Use the properties of eponents to write each epression as a single radical. Then evaluate each epression. Use a calculator to check our answers. a. 1 b. d. 98 e. c. 10 f. 7 Communicate Your Answer. How can ou use properties of eponents to simplif products and quotients of radicals?. Simplif each epression. a. 7 b. 0 1 c. ( 1/ 11/ ) Section. Properties of Rational Eponents and Radicals 199
. Lesson What You Will Learn Core Vocabular simplest form of a radical, p. 01 conjugate, p. 0 like radicals, p. 0 Previous properties of integer eponents rationalizing the denominator absolute value COMMON ERROR When ou multipl powers, do not multipl the eponents. For eample, 10. Use properties of rational eponents to simplif epressions with rational eponents. Use properties of radicals to simplif and write radical epressions in simplest form. Properties of Rational Eponents The properties of integer eponents that ou have previousl learned can also be applied to rational eponents. Core Concept Properties of Rational Eponents Let a and b be real numbers and let m and n be rational numbers, such that the quantities in each propert are real numbers. Propert Name Definition Eample Product of Powers Power of a Power Power of a Product a m a n = a m + n (a m ) n = a mn (ab) m = a m b m 1/ / = (1/ + /) = = ( / ) = (/ ) = = Negative Eponent Zero Eponent Quotient of Powers Power of a Quotient a m = 1 a m, a 0 a 0 = 1, a 0 a m a n = am n, a 0 ( a b m, b 0 b) m = am (1 9) 1/ = 1 1/ 9 1/ = = 1 1/ = 1 0 = 1 1 1/ = 1 / 1/ = (/ 1/) = = 1 ( 7 1/ = ) 71/ 1/ = Using Properties of Eponents Use the properties of rational eponents to simplif each epression. a. 7 1/ 7 1/ = 7 (1/ + 1/) = 7 / b. ( 1/ 1/ ) = ( 1/ ) ( 1/ ) = (1/ ) (1/ ) = 1 / = / c. ( ) 1/ = [( ) ] 1/ = (1 ) 1/ = 1 [ ( 1/)] = 1 1 = 1 1 d. 1 = 1/ ) 1/ e. ( 1/ 1/ = (1 1/) = / = [ ( 1/ ) Monitoring Progress Simplif the epression. ] = (7 1/ ) = 7 (1/ ) = 7 / 1. / 1/. Help in English and Spanish at BigIdeasMath.com 1/. ( 01/ 1/ ). ( 1/ 7 1/ ) 00 Chapter Rational Eponents and Radical Functions
Simplifing Radical Epressions The Power of a Product and Power of a Quotient properties can be epressed using radical notation when m = 1/n for some integer n greater than 1. Core Concept Properties of Radicals Let a and b be real numbers and let n be an integer greater than 1. Propert Name Definition Eample Product Propert Quotient Propert n a b = n a n b n a a n b =, b 0 n b = 8 = 1 = 1 = 81 = Using Properties of Radicals Use the properties of radicals to simplif each epression. a. 1 18 = 1 18 = 1 = Product Propert of Radicals b. 80 = 80 = 1 = Quotient Propert of Radicals An epression involving a radical with inde n is in simplest form when these three conditions are met. No radicands have perfect nth powers as factors other than 1. No radicands contain fractions. No radicals appear in the denominator of a fraction. To meet the last two conditions, rationalize the denominator b multipling the epression b an appropriate form of 1 that eliminates the radical from the denominator. Write each epression in simplest form. Writing Radicals in Simplest Form a. 1 b. 7 8 a. 1 = 7 Factor out perfect cube. = 7 Product Propert of Radicals = Simplif. 7 7 b. = 8 8 8 = 8 = Make the radicand in the denominator a perfect fifth power. Product Propert of Radicals Simplif. Section. Properties of Rational Eponents and Radicals 01
For a denominator that is a sum or difference involving square roots, multipl both the numerator and denominator b the conjugate of the denominator. The epressions a b + c d and a b c d are conjugates of each other, where a, b, c, and d are rational numbers. 1 Write in simplest form. + 1 1 = + + Writing a Radical Epression in Simplest Form The conjugate of + is. = 1 ( ) ( ) Sum and Difference Pattern = Simplif. Radical epressions with the same inde and radicand are like radicals. To add or subtract like radicals, use the Distributive Propert. Adding and Subtracting Like Radicals and Roots Simplif each epression. a. 10 + 7 10 b. (8 1/ ) + 10(8 1/ ) c. a. 10 + 7 10 = (1 + 7) 10 = 8 10 b. (8 1/ ) + 10(8 1/ ) = ( + 10)(8 1/ ) = 1(8 1/ ) c. = 7 = = ( 1) = Monitoring Progress Simplif the epression.. 7. 9. 0 Help in English and Spanish at BigIdeasMath.com 7. 10 8. 10. 7 1 1 11. (9 / ) + 8(9 / ) 1. + 0 The properties of rational eponents and radicals can also be applied to epressions involving variables. Because a variable can be positive, negative, or zero, sometimes absolute value is needed when simplifing a variable epression. Rule Eample 0 Chapter Rational Eponents and Radical Functions When n is odd n n = 7 7 = and 7 ( ) 7 = When n is even n n = = and ( ) = Absolute value is not needed when all variables are assumed to be positive.
Simplifing Variable Epressions STUDY TIP You do not need to take the absolute value of because is being squared. Simplif each epression. a. b. 8 a. = ( ) = ( ) = b. 8 = 8 = ( ) = Writing Variable Epressions in Simplest Form COMMON ERROR You must multipl both the numerator and denominator of the fraction b so that the value of the fraction does not change. Write each epression in simplest form. Assume all variables are positive. a. a 8 b 1 c b. a. a 8 b 1 c = a a b 10 b c b. = 8 8 c. 11/ / z Factor out perfect fifth powers. = a b 10 c a b Product Propert of Radicals = ab c a b Simplif. 8 = 9 = c. 11/ / z = 7 (1 /) 1/ z ( ) = 7 1/ 1/ z Make denominator a perfect cube. Product Propert of Radicals Simplif. Adding and Subtracting Variable Epressions Perform each indicated operation. Assume all variables are positive. a. + b. 1 z z z a. + = ( + ) = 11 b. 1 z z z = 1z z z z = (1z z) z = 9z z Monitoring Progress Help in English and Spanish at BigIdeasMath.com Simplif the epression. Assume all variables are positive. 1. 7q 9 1. 10 1. / 1/ 1/ 1. 9w w w Section. Properties of Rational Eponents and Radicals 0
. Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check 1. WRITING How do ou know when a radical epression is in simplest form?. WHICH ONE DOESN T BELONG? Which radical epression does not belong with the other three? Eplain our reasoning. 11 9 Monitoring Progress and Modeling with Mathematics In Eercises 1, use the properties of rational eponents to simplif the epression. (See Eample 1.). (9 ) 1/. (1 ) 1/. 1/. 7 7 1/ 7. ( 8 1/ 10 ) 8. ( 9 1/ ) 9. ( / 1/ ) 1 10. ( 1/ / ) 1/ 11. / 1 / / 1. 9 /8 9 7/8 7 / In Eercises 1 0, use the properties of radicals to simplif the epression. (See Eample.) 1. 7 1. 1 In Eercises 9, write the epression in simplest form. (See Eample.) 9. 1... 1 1 + 9 + 7 0.... 1 + 11 9 8 + 7 7 10 In Eercises 7, simplif the epression. (See Eample.) 7. 9 11 + 11 8. 8 1 1. 8 1. 8 8 9. (11 1/ ) + 9(11 1/ ) 0. 1(8 / ) (8 / ) 17. 19. 8 7 18. 0. 18 In Eercises 1 8, write the epression in simplest form. (See Eample.) 1.. 7.. 8 7. 9. 88 7. 7 8. 19 1. 1 19. 7 + 7 10. + 7. 7 18. ( 1/ ) ( 1/ ). 1/ + (0 1/ ) 7. ERROR ANALYSIS Describe and correct the error in simplifing the epression. 1 + 1 = ( + ) = 8 = 8 8 = 8 = 1 0 Chapter Rational Eponents and Radical Functions
8. MULTIPLE REPRESENTATIONS Which radical epressions are like radicals? A ( /9 ) / B C E D ( ) 8 1 87 + F 7 80 0 In Eercises 9, simplif the epression. (See Eample.) 9. 81 8 0. r t m 10 1. k n. 1 1z g. h h 8 n 7. 18 p 7 n p 1. ERROR ANALYSIS Describe and correct the error in simplifing the epression. h 1 g h = 1 g = (h ) g = h g In Eercises 70, perform the indicated operation. Assume all variables are positive. (See Eample 8.). 1 + 9. 11 z z 7. 7/ 7/ 8. 7 m 7 + m 7/ 9. 1w 10 + w w 70. (p 1/ p 1/ ) 1p MATHEMATICAL CONNECTIONS In Eercises 71 and 7, find simplified epressions for the perimeter and area of the given figure. 71. / 7. 1/ 1/ 7. MODELING WITH MATHEMATICS The optimum diameter d (in millimeters) of the pinhole in a pinhole camera can be modeled b d = 1.9[(. 10 ) ] 1/, where is the length (in millimeters) of the camera bo. Find the optimum pinhole diameter for a camera bo with a length of 10 centimeters. pinhole film. OPEN-ENDED Write two variable epressions involving radicals, one that needs absolute value in simplifing and one that does not need absolute value. Justif our answers. In Eercises 7, write the epression in simplest form. Assume all variables are positive. (See Eample 7.) 7. 81a 7 b 1 c 9 8. 1r s 9 t 7 10m 9. n 0 7 0. 1 1.. w w w 1. v 7 v 18w 1/ v / 7w / v 1/. 7 / / z / 1/ 1/ tree 7. MODELING WITH MATHEMATICS The surface area S (in square centimeters) of a mammal can be modeled b S = km /, where m is the mass (in grams) of the mammal and k is a constant. The table shows the values of k for different mammals. Mammal Rabbit Human Bat Value of k 9.7 11.0 7. a. Find the surface area of a bat whose mass is grams. b. Find the surface area of a rabbit whose mass is. kilograms (. 10 grams). c. Which mammal has the greatest mass per square centimeter of surface area, the bat in part (a), the rabbit in part (b), or a human whose mass is 9 kilograms? Section. Properties of Rational Eponents and Radicals 0
7. MAKING AN ARGUMENT Your friend claims it is not possible to simplif the epression 7 11 9 because it does not contain like radicals. Is our friend correct? Eplain our reasoning. 7. PROBLEM SOLVING The apparent magnitude of a star is a number that indicates how faint the star is in relation to other stars. The epression.1m1 tells m.1 how man times fainter a star with apparent magnitude m 1 is than a star with apparent magnitude m. Star Apparent magnitude Constellation Vega 0.0 Lra Altair 0.77 Aquila Deneb 1. Cgnus a. How man times fainter is Altair than Vega? b. How man times fainter is Deneb than Altair? c. How man times fainter is Deneb than Vega? Deneb Cgnus Altair Vega Lra Aquila 77. CRITICAL THINKING Find a radical epression for the perimeter of the triangle inscribed in the square shown. Simplif the epression. 78. HOW DO YOU SEE IT? Without finding points, match A. the functions f() = and g() = with their graphs. Eplain our reasoning. 1 1 8 79. REWRITING A FORMULA You have filled two round balloons with water. One balloon contains twice as much water as the other balloon. B. 1 1 8 a. Solve the formula for the volume of a sphere, V = πr, for r. b. Substitute the epression for r from part (a) into the formula for the surface area of a sphere, S = πr. Simplif to show that S = (π) 1/ (V) /. c. Compare the surface areas of the two water balloons using the formula in part (b). 80. THOUGHT PROVOKING Determine whether the epressions ( ) 1/ and ( 1/ ) are equivalent for all values of. 81. DRAWING CONCLUSIONS Substitute different combinations of odd and even positive integers for m and n in the epression n m. When ou cannot assume is positive, eplain when absolute value is needed in simplifing the epression. 8. REWRITING A FORMULA Rewrite the formula in Eercise 7 so that one side is m. Use this formula to S justif our answer in part (c). 8 Maintaining Mathematical Proficienc Reviewing what ou learned in previous grades and lessons Graph the function. Label the verte, ais of smmetr, and -intercepts. (Section.) 8. g() = ( ) 8. h() = ( ) 8. f () = + + Write a rule for g. Describe the graph of g as a transformation of the graph of f. (Section.7) 8. f() =, g() = f() 87. f() =, g() = f() 88. f() =, g() = f( ) 89. f() = +, g() = f() 0 Chapter Rational Eponents and Radical Functions
. Graphing Radical Functions Essential Question Essential Question How can ou identif the domain and range of a radical function? Identifing Graphs of Radical Functions Work with a partner. Match each function with its graph. Eplain our reasoning. Then identif the domain and range of each function. a. f() = b. f() = c. f() = d. f() = A. B. C. D. Identifing Graphs of Transformations Work with a partner. Match each transformation of f() = with its graph. Eplain our reasoning. Then identif the domain and range of each function. a. g() = + b. g() = c. g() = + d. g() = + A. B. C. D. LOOKING FOR STRUCTURE To be proficient in math, ou need to look closel to discern a pattern or structure. Communicate Your Answer. How can ou identif the domain and range of a radical function?. Use the results of Eploration 1 to describe how the domain and range of a radical function are related to the inde of the radical. Section. Graphing Radical Functions 07
. Lesson What You Will Learn Core Vocabular radical function, p. 08 Previous transformations parabola circle STUDY TIP A power function has the form = a b, where a is a real number and b is a rational number. Notice that the parent square root function is a power function, where a = 1 and b = 1. Graph radical functions. Write transformations of radical functions. Graph parabolas and circles. Graphing Radical Functions A radical function contains a radical epression with the independent variable in the radicand. When the radical is a square root, the function is called a square root function. When the radical is a cube root, the function is called a cube root function. Core Concept Parent Functions for Square Root and Cube Root Functions The parent function for the famil of The parent function for the famil of square root functions is f() =. cube root functions is f() =. (0, 0) f() = (1, 1) Domain: 0, Range: 0 (0, 0) ( 1, 1) f() = (1, 1) Domain and range: All real numbers Graphing Radical Functions Graph each function. Identif the domain and range of each function. LOOKING FOR STRUCTURE Eample 1(a) uses -values that are multiples of so that the radicand is an integer. a. f() = 1 b. g() = a. Make a table of values and sketch the graph. 0 8 1 1 f() = 1 0 1 1.1 1.7 1 The radicand of a square root must be nonnegative. So, the domain is 0. The range is 0. b. Make a table of values and sketch the graph. 8 1 1 1 0 1 g() =.78 0.78 The radicand of a cube root can be an real number. So, the domain and range are all real numbers. 08 Chapter Rational Eponents and Radical Functions
In Eample 1, notice that the graph of f is a horizontal stretch of the graph of the parent square root function. The graph of g is a vertical stretch and a reflection in the -ais of the graph of the parent cube root function. You can transform graphs of radical functions in the same wa ou transformed graphs of functions previousl. Core Concept Transformation f() Notation Eamples Horizontal Translation g() = units right f( h) Graph shifts left or right. g() = + units left Vertical Translation Graph shifts up or down. f() + k g() = + 7 g() = 1 7 units up 1 unit down Reflection Graph flips over - or -ais. f( ) f() g() = g() = in the -ais in the -ais Horizontal Stretch or Shrink Graph stretches awa from or shrinks toward -ais. Vertical Stretch or Shrink Graph stretches awa from or shrinks toward -ais. f(a) a f() g() = shrink b a g() = 1 factor of 1 stretch b a factor of g() = stretch b a factor of g() = 1 shrink b a factor of 1 Transforming Radical Functions Describe the transformation of f represented b g. Then graph each function. a. f() =, g() = + b. f() =, g() = 8 LOOKING FOR STRUCTURE In Eample (b), ou can use the Product Propert of Radicals to write g() =. So, ou can also describe the graph of g as a vertical stretch b a factor of and a reflection in the -ais of the graph of f. a. Notice that the function is of the form g() = h + k, where h = and k =. So, the graph of g is a translation units right and units up of the graph of f. g f b. Notice that the function is of the form g() = a, where a = 8. So, the graph of g is a horizontal shrink b a factor of 1 8 and a reflection in the -ais of the graph of f. f g Monitoring Progress Help in English and Spanish at BigIdeasMath.com 1. Graph g() = + 1. Identif the domain and range of the function.. Describe the transformation of f () = represented b g() =. Then graph each function. Section. Graphing Radical Functions 09
Writing Transformations of Radical Functions Modeling with Mathematics The function E(d ) = 0. d approimates the number of seconds it takes a dropped object to fall d feet on Earth. The function M(d ) = 1. E(d ) approimates the number of seconds it takes a dropped object to fall d feet on Mars. Write a rule for M. How long does it take a dropped object to fall feet on Mars? Self-Portrait of NASA s Mars Rover Curiosit 1. Understand the Problem You are given a function that represents the number of seconds it takes a dropped object to fall d feet on Earth. You are asked to write a similar function for Mars and then evaluate the function for a given input.. Make a Plan Multipl E(d ) b 1. to write a rule for M. Then find M().. Solve the Problem M(d) = 1. E(d ) Net, find M(). = 1. 0. d Substitute 0. d for E(d ). = 0. d Simplif. M() = 0. = 0.(8) =. It takes a dropped object about. seconds to fall feet on Mars.. Look Back Use the original functions to check our solution. E() = 0. = M() = 1. E() = 1. =. Writing a Transformed Radical Function Let the graph of g be a horizontal shrink b a factor of 1 followed b a translation units to the left of the graph of f() =. Write a rule for g. Step 1 First write a function h that represents the horizontal shrink of f. Check 7 g h f h() = f() Multipl the input b 1 1 =. = Replace with in f(). Step Then write a function g that represents the translation of h. g() = h( + ) Subtract, or add, to the input. = ( + ) Replace with + in h(). = + 18 Distributive Propert The transformed function is g() = + 18. Monitoring Progress Help in English and Spanish at BigIdeasMath.com. WHAT IF? In Eample, the function N(d ) =. E(d ) approimates the number of seconds it takes a dropped object to fall d feet on the Moon. Write a rule for N. How long does it take a dropped object to fall feet on the Moon?. In Eample, is the transformed function the same when ou perform the translation followed b the horizontal shrink? Eplain our reasoning. 10 Chapter Rational Eponents and Radical Functions
Graphing Parabolas and Circles To graph parabolas and circles using a graphing calculator, first solve their equations for to obtain radical functions. Then graph the functions. Graphing a Parabola (Horizontal Ais of Smmetr) Use a graphing calculator to graph 1 =. Identif the verte and the direction that the parabola opens. STUDY TIP Notice 1 is a function and is a function, but 1 = is not a function. Step 1 Solve for. 1 = Write the original equation. = Multipl each side b. = ± Step Graph both radical functions. 1 = Take square root of each side. 1 = 10 The verte is (0, 0) and the parabola opens right. Graphing a Circle Use a graphing calculator to graph + = 1. Identif the center, radius, and intercepts. Step 1 Solve for. + = 1 = 1 Write the original equation. Subtract from each side. = ± 1 Take square root of each side. Step Graph both radical functions using a square viewing window. 9 1 9 1 = 1 = 1 The center is (0, 0) and the radius is units. The -intercepts are ±. The -intercepts are also ±. Monitoring Progress Help in English and Spanish at BigIdeasMath.com. Use a graphing calculator to graph = + 1. Identif the verte and the direction that the parabola opens.. Use a graphing calculator to graph ( + ) + =. Identif the center, radius, and intercepts. Section. Graphing Radical Functions 11
. Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check 1. COMPLETE THE SENTENCE Square root functions and cube root functions are eamples of functions.. COMPLETE THE SENTENCE When graphing = a h + k, translate the graph of = a h units and k units. Monitoring Progress and Modeling with Mathematics In Eercises 8, match the function with its graph.. f() = +. h() = +. f() =. g() = 7. h() = + 8. f() = + In Eercises 19, describe the transformation of f represented b g. Then graph each function. (See Eample.) 19. f() =, g() = + 1 + 8 0. f() =, g() = 1 A. B. 1. f() =, g() = 1. f() =, g() = +. f() = 1/, g() = 1 ( )1/ C. D.. f() = 1/, g() = 1 1/ +. f() =, g() = +. f() =, g() = + E. F. In Eercises 9 18, graph the function. Identif the domain and range of the function. (See Eample 1.) 9. h() = + 10. g() = 11. g() = 1. f() = 1. g() = 1 1. f() = 1 + 1. f() = () 1/ + 1. g() = ( + 1) 1/ 17. h() = 18. h() = 7. ERROR ANALYSIS Describe and correct the error in graphing f() =. 8. ERROR ANALYSIS Describe and correct the error in describing the transformation of the parent square root function represented b g() = 1 +. The graph of g is a horizontal shrink b a factor of 1 and a translation units up of the parent square root function. 1 Chapter Rational Eponents and Radical Functions
USING TOOLS In Eercises 9, use a graphing calculator to graph the function. Then identif the domain and range of the function. 9. g() = + 0. h() = 1. f() = +. f() =. f() = + + 1. h() = 1 + ABSTRACT REASONING In Eercises 8, complete the statement with sometimes, alwas, or never.. The domain of the function = a is 0.. The range of the function = a is 0. 7. The domain and range of the function = h + k are all real numbers. 8. The domain of the function = a + k is 0. 9. PROBLEM SOLVING The distance (in miles) a pilot can see to the horizon can be approimated b E(n) = 1. n, where n is the plane s altitude (in feet above sea level) on Earth. The function M(n) = 0.7E(n) approimates the distance a pilot can see to the horizon n feet above the surface of Mars. Write a rule for M. What is the distance a pilot can see to the horizon from an altitude of 10,000 feet above Mars? (See Eample.) n 0. MODELING WITH MATHEMATICS The speed (in knots) of sound waves in air can be modeled b v(k) =.8 K 7.1 where K is the air temperature (in kelvin). The speed (in meters per second) of sound waves in air can be modeled b s(k) = v(k) 1.9. Write a rule for s. What is the speed (in meters per second) of sound waves when the air temperature is 0 kelvin? In Eercises 1, write a rule for g described b the transformations of the graph of f. (See Eample.) 1. Let g be a vertical stretch b a factor of, followed b a translation units up of the graph of f() = +.. Let g be a reflection in the -ais, followed b a translation 1 unit right of the graph of f() = 1.. Let g be a horizontal shrink b a factor of, followed b a translation units left of the graph of f() =.. Let g be a translation 1 unit down and units right, followed b a reflection in the -ais of the graph of 1 f() = + In Eercises and, write a rule for g.. g f() =.. f() = g In Eercises 7 0, write a rule for g that represents the indicated transformation of the graph of f. 7. f() =, g() = f( + ) 8. f() = 1 1, g() = f() + 9 9. f() =, g() = f( + ) 0. f() = + 10, g() = 1 f( ) + In Eercises 1, use a graphing calculator to graph the equation of the parabola. Identif the verte and the direction that the parabola opens. (See Eample.) 1. 1 =. =. 8 + =. =. + 8 = 1. 1 = In Eercises 7, use a graphing calculator to graph the equation of the circle. Identif the center, radius, and intercepts. (See Eample.) 7. + = 9 8. + = 9. ( 1) = 0. 1 ( + ) = 1. + + 1 1 = 0. + + = 9 Section. Graphing Radical Functions 1
. MODELING WITH MATHEMATICS The period of a pendulum is the time the pendulum takes to complete one back-and-forth swing. The period T (in seconds) can be modeled b the function T = 1.11, where is the length (in feet) of the pendulum. Graph the function. Estimate the length of a pendulum with a period of seconds. Eplain our reasoning.. HOW DO YOU SEE IT? Does the graph represent a square root function or a cube root function? Eplain. What are the domain and range of the function? (, ) (, 1). PROBLEM SOLVING For a drag race car with a total weight of 00 pounds, the speed s (in miles per hour) at the end of a race can be modeled b s = 1.8 p, where p is the power (in horsepower). Graph the function. a. Determine the power of a 00-pound car that reaches a speed of 00 miles per hour. b. What is the average rate of change in speed as the power changes from 1000 horsepower to 100 horsepower?. THOUGHT PROVOKING The graph of a radical function f passes through the points (, 1) and (, 0). Write two different functions that could represent f( + ) + 1. Eplain. 7. MULTIPLE REPRESENTATIONS The terminal velocit v t (in feet per second) of a skdiver who weighs 10 pounds is given b v t =.7 10 A where A is the cross-sectional surface area (in square feet) of the skdiver. The table shows the terminal velocities (in feet per second) for various surface areas (in square feet) of a skdiver who weighs 1 pounds. Cross-sectional surface area, A Terminal velocit, v t 1.9 9.9 19. 7 1. a. Which skdiver has a greater terminal velocit for each value of A given in the table? b. Describe how the different values of A given in the table relate to the possible positions of the falling skdiver. 8. MATHEMATICAL CONNECTIONS The surface area S of a right circular cone with a slant height of 1 unit is given b S = πr + πr, where r is the radius of the cone. r 1 unit a. Use completing the square to show that r = 1 π S + π 1. b. Graph the equation in part (a) using a graphing calculator. Then find the radius of a right circular cone with a slant height of 1 unit and a surface area of π square units. Maintaining Mathematical Proficienc Reviewing what ou learned in previous grades and lessons Solve the equation. Check our solutions. (Skills Review Handbook) 9. + = 70. + 9 = 7 71. = 7. + 8 = + Solve the inequalit. Graph the solution on a number line. (Skills Review Handbook) 7. > 8 7. + 7 7. ( ) 1 Chapter Rational Eponents and Radical Functions
.1. What Did You Learn? Core Vocabular nth root of a, p. 19 inde of a radical, p. 19 simplest form of a radical, p. 01 conjugate, p. 0 like radicals, p. 0 radical function, p. 08 Core Concepts Section.1 Real nth Roots of a, p. 19 Rational Eponents, p. 19 Section. Properties of Rational Eponents, p. 00 Properties of Radicals, p. 01 Section. Parent Functions for Square Root and Cube Root Functions, p. 08 Transformations of Radical Functions, p. 09 Mathematical Practices 1. How can ou use definitions to eplain our reasoning in Eercises 1 on page 197?. How did ou use structure to solve Eercise 7 on page 0?. How can ou check that our answer is reasonable in Eercise 9 on page 1?. How can ou make sense of the terms of the surface area formula given in Eercise 8 on page 1? Analzing Your Errors Application Errors What Happens: You can do numerical problems, but ou struggle with problems that have contet. How to Avoid This Error: Do not just mimic the steps of solving an application problem. Eplain out loud what the question is asking and wh ou are doing each step. After solving the problem, ask ourself, Does m solution make sense? 1
.1. Quiz Find the indicated real nth root(s) of a. (Section.1) 1. n =, a = 81. n =, a = 10. Evaluate (a) 1 / and (b) 1 / without using a calculator. Eplain our reasoning. (Section.1) Find the real solution(s) of the equation. Round our answer to two decimal places when appropriate. (Section.1). = 18. ( + ) = 8 Simplif the epression. (Section.). ( 81/ 1/ ) 7. 8. 1 + 9. 1 10. Simplif 8 9 8 z 1. (Section.) Write the epression in simplest form. Assume all variables are positive. (Section.) 11. 1p 9 1. m 1. Graph f() = + 1. Identif the domain and range of the function. (Section.) 1. n q + 7n q Describe the transformation of the graph of f represented b the graph of g. Then write a rule for g. (Section.) 1. f() = 1. f() = 17. f() = g g 1 1 1 g 18. Use a graphing calculator to graph =. Identif the verte and direction the parabola opens. (Section.) 19. A jeweler is setting a stone cut in the shape of a regular octahedron. A regular octahedron is a solid with eight equilateral triangles as faces, as shown. The formula for the volume of the stone is V = 0.7s, where s is the side length (in millimeters) of an edge of the stone. The volume of the stone is 11 cubic millimeters. Find the length of an edge of the stone. (Section.1) 0. An investigator can determine how fast a car was traveling just prior to an accident using the model s = d, where s is the speed (in miles per hour) of the car and d is the length (in feet) of the skid marks. Graph the model. The length of the skid marks of a car is 90 feet. Was the car traveling at the posted speed limit prior to the accident? Eplain our reasoning. (Section.) s SPEED LIMIT 1 Chapter Rational Eponents and Radical Functions
. Solving Radical Equations and Inequalities Essential Question How can ou solve a radical equation? Solving Radical Equations Work with a partner. Match each radical equation with the graph of its related radical function. Eplain our reasoning. Then use the graph to solve the equation, if possible. Check our solutions. a. 1 1 = 0 b. + + = 0 c. 9 = 0 d. + = 0 e. + = 0 f. + 1 = 0 A. B. C. D. E. F. LOOKING FOR STRUCTURE To be proficient in math, ou need to look closel to discern a pattern or structure. Solving Radical Equations Work with a partner. Look back at the radical equations in Eploration 1. Suppose that ou did not know how to solve the equations using a graphical approach. a. Show how ou could use a numerical approach to solve one of the equations. For instance, ou might use a spreadsheet to create a table of values. b. Show how ou could use an analtical approach to solve one of the equations. For instance, look at the similarities between the equations in Eploration 1. What first step ma be necessar so ou could square each side to eliminate the radical(s)? How would ou proceed to find the solution? Communicate Your Answer. How can ou solve a radical equation?. Would ou prefer to use a graphical, numerical, or analtical approach to solve the given equation? Eplain our reasoning. Then solve the equation. + = 1 Section. Solving Radical Equations and Inequalities 17
. Lesson What You Will Learn Core Vocabular radical equation, p. 18 etraneous solutions, p. 19 Previous rational eponents radical epressions solving quadratic equations Solve equations containing radicals and rational eponents. Solve radical inequalities. Solving Equations Equations with radicals that have variables in their radicands are called radical equations. An eample of a radical equation is + 1 =. Core Concept Solving Radical Equations To solve a radical equation, follow these steps: Step 1 Isolate the radical on one side of the equation, if necessar. Step Raise each side of the equation to the same eponent to eliminate the radical and obtain a linear, quadratic, or other polnomial equation. Step Solve the resulting equation using techniques ou learned previousl. Check our solution. Solving Radical Equations Solve (a) + 1 = and (b) 9 1 =. a. + 1 = Write the original equation. Check + 1 =? =? = + 1 = Divide each side b. ( + 1 ) = Square each side to eliminate the radical. + 1 = Simplif. = Subtract 1 from each side. b. The solution is =. 9 1 = 9 = Write the original equation. Add 1 to each side. Check (18) 9 1 =? 7 1 =? = ( 9 ) = Cube each side to eliminate the radical. 9 = 7 Simplif. = Add 9 to each side. = 18 Divide each side b. The solution is = 18. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Solve the equation. Check our solution. 1. 9 =. + =. = 18 Chapter Rational Eponents and Radical Functions
Solving a Real-Life Problem In a hurricane, the mean sustained wind velocit v (in meters per second) can be modeled b v( p) =. 101 p, where p is the air pressure (in millibars) at the center of the hurricane. Estimate the air pressure at the center of the hurricane when the mean sustained wind velocit is. meters per second. v( p) =. 101 p Write the original function.. =. 101 p Substitute. for v( p). 8. 101 p Divide each side b.. 8. ( 101 p ) Square each side. 7.8 101 p Simplif. 98. p Subtract 101 from each side. ATTEND TO PRECISION To understand how etraneous solutions can be introduced, consider the equation =. This equation has no real solution; however, ou obtain = 9 after squaring each side. 98. p Divide each side b 1. The air pressure at the center of the hurricane is about 98 millibars. Monitoring Progress Help in English and Spanish at BigIdeasMath.com. WHAT IF? Estimate the air pressure at the center of the hurricane when the mean sustained wind velocit is 8. meters per second. Raising each side of an equation to the same eponent ma introduce solutions that are not solutions of the original equation. These solutions are called etraneous solutions. When ou use this procedure, ou should alwas check each apparent solution in the original equation. Solve + 1 = 7 + 1. Solving an Equation with an Etraneous Solution + 1 = 7 + 1 Write the original equation. ( + 1) = ( 7 + 1 ) Square each side. + + 1 = 7 + 1 1 = 0 Epand left side and simplif right side. Write in standard form. ( 7)( + ) = 0 Factor. 7 = 0 or + = 0 Zero-Product Propert = 7 or = Solve for. Check 7 + 1 =? 7(7) + 1 + 1 =? 7( ) + 1 8 =? 1 =? 1 8 = 8 1 1 The apparent solution = is etraneous. So, the onl solution is = 7. Section. Solving Radical Equations and Inequalities 19
Solve + + 1 =. Solving an Equation with Two Radicals + + 1 = Write the original equation. ( + + 1 ) = ( ) Square each side. + + + + 1 = Epand left side and simplif right side. + = Isolate radical epression. + = Divide each side b. ( + ) = ( ) Square each side. ANOTHER METHOD You can also graph each side of the equation and find the -value where the graphs intersect. + = Simplif. 0 = Write in standard form. 0 = ( )( + 1) Factor. = 0 or + 1 = 0 Zero-Product Propert = or = 1 Solve for. Intersection X=-1 Y= Check + + 1 =? 1 + + 1 =? ( 1) + 1 =? 1 1 + 1 =? 1 = The apparent solution = is etraneous. So, the onl solution is = 1. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Solve the equation. Check our solution(s).. 10 + 9 = +. + = + 7 7. + = When an equation contains a power with a rational eponent, ou can solve the equation using a procedure similar to the one for solving radical equations. In this case, ou first isolate the power and then raise each side of the equation to the reciprocal of the rational eponent. Solve () / + = 10. Solving an Equation with a Rational Eponent () / + = 10 () / = 8 [() / ] / = 8 / Write the original equation. Subtract from each side. Raise each side to the four-thirds. Check = 1 Simplif. = 8 Divide each side b. The solution is = 8. ( 8) / + =? 10 1 / + =? 10 10 = 10 0 Chapter Rational Eponents and Radical Functions
Solve ( + 0) 1/ =. Solving an Equation with a Rational Eponent Check ( + 0) 1/ =? 1/ =? = ( + 0) 1/ =? 1/ =? ( + 0) 1/ = Write the original equation. [( + 0) 1/ ] = Square each side. + 0 = Simplif. 0 = 0 Write in standard form. 0 = ( )( + ) Factor. = 0 or + = 0 Zero-Product Propert = or = Solve for. The apparent solution = is etraneous. So, the onl solution is =. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Solve the equation. Check our solution(s). 8. () 1/ = 9. ( + ) 1/ = 10. ( + ) / = 8 Solving Radical Inequalities To solve a simple radical inequalit of the form n u < d, where u is an algebraic epression and d is a nonnegative number, raise each side to the eponent n. This procedure also works for >,, and. Be sure to consider the possible values of the radicand. Solving a Radical Inequalit Solve 1 1. Check 0 = 1 Step 1 Solve for. 1 1 Write the original inequalit. 1 Divide each side b. 1 1 Square each side. 17 Add 1 to each side. Intersection X=17 Y=1 8 = 1 Step Consider the radicand. 1 0 1 So, the solution is 1 17. The radicand cannot be negative. Add 1 to each side. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 11. Solve (a) and (b) + 1 < 8. Section. Solving Radical Equations and Inequalities 1
. Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check 1. VOCABULARY Is the equation = a radical equation? Eplain our reasoning.. WRITING Eplain the steps ou should use to solve + 10 < 1. Monitoring Progress and Modeling with Mathematics In Eercises 1, solve the equation. Check our solution. (See Eample 1.). + 1 =. + 10 = 8 In Eercises 1, solve the equation. Check our solution(s). (See Eamples and.) 1. = 1. 10 = 9. 1 =. 10 = 7 17. = 10 18. + 0 = + 7. + 1 = 11 19. 8 1 = 1 0. 8 = 8. 8 10 1 = 17 1. + 1 = + 10. + 1 = 0 9. 1 + 10 = 8 10. = 0 11. + 7 = 1 1. 1 = 1.. 8 + 1 = 0 + = +. 8 + 1 = + 1. MODELING WITH MATHEMATICS Biologists have discovered that the shoulder height h (in centimeters) of a male Asian elephant can be modeled b h =. t + 7.8, where t is the age (in ears) of the elephant. Determine the age of an elephant with a shoulder height of 0 centimeters. (See Eample.). + = In Eercises 7, solve the equation. Check our solution(s). (See Eamples and.) 7. / = 8 8. / = 9. 1/ + = 0 0. / 1 = 0 1. ( + ) 1/ =. ( ) 1/ = 0 h. ( + 11) 1/ = +. ( ) 1/ = ERROR ANALYSIS In Eercises and, describe and correct the error in solving the equation. 1. MODELING WITH MATHEMATICS In an amusement park ride, a rider suspended b cables swings back and forth from a tower. The maimum speed v (in meters per second) of the rider can be approimated b v = gh, where h is the height (in meters) at the top of each swing and g is the acceleration due to gravit (g 9.8 m/sec ). Determine the height at the top of the swing of a rider whose maimum speed is 1 meters per second.. 8 = ( 8 ) = 8 = = 1 =. 8 / = 1000 8( / ) / = 1000 / 8 = 100 = Chapter Rational Eponents and Radical Functions
In Eercises 7, solve the inequalit. (See Eample 7.) 7. 8. 9. > 0 0. 7 + 1 < 9 1. + 8. + 7. + < 1. 0.. MODELING WITH MATHEMATICS The length (in inches) of a standard nail can be modeled b = d /, where d is the diameter (in inches) of the nail. What is the diameter of a standard nail that is inches long?. DRAWING CONCLUSIONS Hang time is the time ou are suspended in the air during a jump. Your hang time t (in seconds) is given b the function t = 0. h, where h is the height (in feet) of the jump. Suppose a kangaroo and a snowboarder jump with the hang times shown. t = 0.81 t = 1.1 a. Find the heights that the snowboarder and the kangaroo jump. b. Double the hang times of the snowboarder and the kangaroo and calculate the corresponding heights of each jump. c. When the hang time doubles, does the height of the jump double? Eplain. In Eercises 7, solve the nonlinear sstem. Justif our answer with a graph. 7. = 8. = + 17 = = + 9. + = 0. + = = = + 1. + = 1. + = = 1 1 = +. PROBLEM SOLVING The speed s (in miles per hour) of a car can be given b s = 0 fd, where f is the coefficient of friction and d is the stopping distance (in feet). The table shows the coefficient of friction for different surfaces. Surface Coefficient of friction, f dr asphalt 0.7 wet asphalt 0.0 snow 0.0 ice 0.1 a. Compare the stopping distances of a car traveling miles per hour on the surfaces given in the table. b. You are driving miles per hour on an ic road when a deer jumps in front of our car. How far awa must ou begin to brake to avoid hitting the deer? Justif our answer.. MODELING WITH MATHEMATICS The Beaufort wind scale was devised to measure wind speed. The Beaufort numbers B, which range from 0 to 1, can be modeled b B = 1.9 s +.., where s is the wind speed (in miles per hour). Beaufort number Force of wind 0 calm gentle breeze strong breeze 9 strong gale 1 hurricane a. What is the wind speed for B = 0? B =? b. Write an inequalit that describes the range of wind speeds represented b the Beaufort model.. REASONING Solve the equation =. Then solve the equation =. a. How does changing to change the solution(s) of the equation? b. Justif our answer in part (a) using graphs.. MAKING AN ARGUMENT Your friend sas it is impossible for a radical equation to have two etraneous solutions. Is our friend correct? Eplain our reasoning. Section. Solving Radical Equations and Inequalities
7. USING STRUCTURE Eplain how ou know the radical equation + = has no real solution without solving it. 8. HOW DO YOU SEE IT? Use the graph to find the solution of the equation = 1 +. Eplain our reasoning. = = 1 + (, ) 1. MATHEMATICAL CONNECTIONS The Moeraki Boulders along the coast of New Zealand are stone spheres with radii of approimatel feet. A formula for the radius of a sphere is S r = 1 π where S is the surface area of the sphere. Find the surface area of a Moeraki Boulder.. PROBLEM SOLVING You are tring to determine the height of a truncated pramid, which cannot be measured directl. The height h and slant height of the truncated pramid are related b the formula below. 9. WRITING A compan determines that the price p of a product can be modeled b p = 70 0.0 + 1, where is the number of units of the product demanded per da. Describe the effect that raising the price has on the number of units demanded. 0. THOUGHT PROVOKING Cit officials rope off a circular area to prepare for a concert in the park. The estimate that each person occupies square feet. Describe how ou can use a radical inequalit to determine the possible radius of the region when P people are epected to attend the concert. = h + 1 (b b 1 ) In the given formula, b 1 and b are the side lengths of the upper and lower bases of the pramid, respectivel. When =, b 1 =, and b =, what is the height of the pramid?. REWRITING A FORMULA A burning candle has a radius of r inches and was initiall h 0 inches tall. After t minutes, the height of the candle has been reduced to h inches. These quantities are related b the formula r = kt π (h 0 h) h where k is a constant. Suppose the radius of a candle is 0.87 inch, its initial height is. inches, and k = 0.0. a. Rewrite the formula, solving for h in terms of t. b. Use our formula in part (a) to determine the height of the candle after it burns for minutes. Maintaining Mathematical Proficienc Perform the indicated operation. (Section. and Section.) Reviewing what ou learned in previous grades and lessons. ( + + 1) + ( 7). ( + ) ( ). ( + + 1)( + ) 7. ( + + 11 + 1 1) ( + ) Let f() = +. Write a rule for g. Describe the graph of g as a transformation of the graph of f. (Section.7) 8. g() = f( ) + 9. g() = 1 f() 70. g() = f( 1) + Chapter Rational Eponents and Radical Functions
. Performing Function Operations Essential Question How can ou use the graphs of two functions to sketch the graph of an arithmetic combination of the two functions? Just as two real numbers can be combined b the operations of addition, subtraction, multiplication, and division to form other real numbers, two functions can be combined to form other functions. For eample, the functions f() = and g() = 1 can be combined to form the sum, difference, product, or quotient of f and g. f() + g() = ( ) + ( 1) = + sum f() g() = ( ) ( 1) = + difference f() g() = ( )( 1) = + product f() g() = 1 quotient Graphing the Sum of Two Functions Work with a partner. Use the graphs of f and g to sketch the graph of f + g. Eplain our steps. Sample Choose a point on the = f() graph of g. Use a compass or a 8 ruler to measure its distance above or below the -ais. If above, add the distance to the -coordinate of the point with the same = f() + g() -coordinate on the graph of f. If 8 below, subtract the distance. Plot the new point. Repeat this process = g() for several points. Finall, draw a smooth curve through the new 8 points to obtain the graph of f + g. a. 8 b. 8 = f() = g() MAKING SENSE OF PROBLEMS To be proficient in math, ou need to check our answers to problems using a different method and continuall ask ourself, Does this make sense? 8 = g() 8 8 Communicate Your Answer 8 = f(). How can ou use the graphs of two functions to sketch the graph of an arithmetic combination of the two functions?. Check our answers in Eploration 1 b writing equations for f and g, adding the functions, and graphing the sum. 8 8 Section. Performing Function Operations
. Lesson What You Will Learn Core Vocabular Previous domain scientific notation Add, subtract, multipl, and divide functions. Operations on Functions You have learned how to add, subtract, multipl, and divide polnomial epressions. These operations can also be defined for functions. Core Concept Operations on Functions Let f and g be an two functions. A new function can be defined b performing an of the four basic operations on f and g. Operation Definition Eample: f() =, g() = + Addition ( f + g)() = f() + g() ( f + g)() = + ( + ) = + Subtraction ( f g)() = f() g() ( f g)() = ( + ) = Multiplication ( fg)() = f() g() ( fg)() = ( + ) = + 10 Division ( g) f f() () = g() ( g) f () = + The domains of the sum, difference, product, and quotient functions consist of the -values that are in the domains of both f and g. Additionall, the domain of the quotient does not include -values for which g() = 0. Adding Two Functions Let f() = and g() = 10. Find ( f + g)() and state the domain. Then evaluate the sum when =. ( f + g)() = f() + g() = + ( 10 ) = ( 10) = 7 The functions f and g each have the same domain: all nonnegative real numbers. So, the domain of f + g also consists of all nonnegative real numbers. To evaluate f + g when =, ou can use several methods. Here are two: Method 1 Use an algebraic approach. When =, the value of the sum is ( f + g)() = 7 = 1. Method Use a graphical approach. Enter the functions 1 =, = 10, and = 1 + in a graphing calculator. Then graph, the sum of the two functions. Use the trace feature to find the value of f + g when =. From the graph, ( f + g)() = 1. Y=Y1+Y X= Y=-1 0 8 The value of (f + g)() is 1. Chapter Rational Eponents and Radical Functions
Subtracting Two Functions Let f() = + and g() = +. Find ( f g)() and state the domain. Then evaluate the difference when =. ( f g)() = f() g() = + ( + ) = + + 7 The functions f and g each have the same domain: all real numbers. So, the domain of f g also consists of all real numbers. When =, the value of the difference is ( f g)( ) = ( ) + ( ) ( ) + 7 =. 8 Y=Y1*Y X=0 Y=0 The domain of fg is all nonnegative real numbers. 8 Multipling Two Functions Let f() = and g() =. Find ( fg)() and state the domain. Then evaluate the product when = 9. ( fg)() = f() g() = ( ) = ( 1/ ) = (+1/) = / The domain of f consists of all real numbers, and the domain of g consists of all nonnegative real numbers. So, the domain of fg consists of all nonnegative real numbers. To confirm this, enter the functions 1 =, =, and = 1 in a graphing calculator. Then graph, the product of the two functions. It appears from the graph that the domain of fg consists of all nonnegative real numbers. When = 9, the value of the product is ( fg)(9) = 9 / = (9 1/ ) = =. Dividing Two Functions g) Let f() = and g() = /. Find ( f () and state the domain. Then evaluate the quotient when = 1. ANOTHER WAY In Eample, ou can also g) evaluate ( f (1) as ( g) f (1) = f(1) g(1) = (1) (1) / = 9 8 = 1. ( f f() () = g() = / = (1 /) = 1/ g) The domain of f consists of all real numbers, and the domain of g consists of all nonnegative real numbers. Because g(0) = 0, the domain of f is restricted to all g positive real numbers. When = 1, the value of the quotient is ( f g) (1) = (1)1/ = ( ) 1/ = 1. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 1. Let f() = / and g() = 7 /. Find ( f + g)() and ( f g)() and state the domain of each. Then evaluate ( f + g)(8) and ( f g)(8).. Let f() = and g() = 1/. Find ( fg)() and ( f () and state the domain of each. Then evaluate ( fg)() and ( g) f (). g) Section. Performing Function Operations 7
Performing Function Operations Using Technolog Let f() = and g() = 9. Use a graphing calculator to evaluate ( f + g)(), g) ( f g)(), ( fg)(), and ( f () when =. Round our answers to two decimal places. Enter the functions 1 = and = 9 in a graphing calculator. On the home screen, enter 1 () + (). The first entr on the screen shows that 1 () + ()., so ( f + g)().. Enter the other function operations as shown. Here are the results of the other function operations rounded to two decimal places: ( f g)() 0.8 ( fg)().1 ( f () 0. g) Solving a Real-Life Problem For a white rhino, heart rate r (in beats per minute) and life span s (in minutes) are related to bod mass m (in kilograms) b the functions r(m) = 1m 0. and s(m) = ( 10 )m 0.. a. Find (rs)(m). b. Eplain what (rs)(m) represents. Y1()+Y().081 Y1()-Y() -.81811 Y1()*Y().177 Y1()/Y(). a. (rs)(m) = r(m) s(m) Definition of multiplication = 1m 0. [( 10 )m 0. ] Write product of r (m) and s(m). = 1( 10 )m 0.+0. Product of Powers Propert = (1 10 )m 0.0 Simplif. = (1. 10 9 )m 0.0 Use scientific notation. b. Multipling heart rate b life span gives the total number of heartbeats over the lifetime of a white rhino with bod mass m. Monitoring Progress Help in English and Spanish at BigIdeasMath.com. Let f() = 8 and g() = /. Use a graphing calculator to evaluate ( f + g)(), g) ( f g)(), ( fg)(), and ( f () when =. Round our answers to two decimal places.. In Eample, eplain wh ou can evaluate ( f + g)(), ( f g)(), and ( fg)() but not ( g) f ().. Use the answer in Eample (a) to find the total number of heartbeats over the lifetime of a white rhino when its bod mass is 1.7 10 kilograms. 8 Chapter Rational Eponents and Radical Functions
. Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check 1. WRITING Let f and g be an two functions. Describe how ou can use f, g, and the four basic operations to create new functions.. WRITING What -values are not included in the domain of the quotient of two functions? Monitoring Progress and Modeling with Mathematics In Eercises, find ( f + g)() and ( f g)() and state the domain of each. Then evaluate f + g and f g for the given value of. (See Eamples 1 and.). f() =, g() = 19 ; = 1. f() =, g() = 11 ; =. f() = 7, g() = 9 ; = 1. f() = 11 +, g() = 7 + ; = In Eercises 7 1, find ( fg)() and ( f () and state the domain of each. Then evaluate fg and f for the given g value of. (See Eamples and.) 7. f() =, g() = ; = 7 8. f() =, g() = ; = 9. f() =, g() = 9 1/ ; = 9 10. f() = 11, g() = 7 7/ ; = 8 g) 11. f() = 7 /, g() = 1 1/ ; = 1. f() = /, g() = 1/ ; = 1 USING TOOLS In Eercises 1 1, use a graphing calculator to evaluate ( f + g)(), ( f g)(), ( fg)(), g) and ( f () when =. Round our answers to two decimal places. (See Eample.) 1. f() = ; g() = 1/ 1. f() = 7 / ; g() = 9 / 1. f() = 1/ ; g() = 1/ 1. f() = 1/ ; g() = / ERROR ANALYSIS In Eercises 17 and 18, describe and correct the error in stating the domain. 17. 18. f ( ) = 1/ and g ( ) = / The domain of fg is all real numbers. f ( ) = and g ( ) = The domain of f is all g real numbers ecept =. 19. MODELING WITH MATHEMATICS From 1990 to 010, the numbers (in millions) of female F and male M emploees from the ages of 1 to 19 in the United States can be modeled b F(t) = 0.007t + 0.10t +.7 and M(t) = 0.0001t 0.009t + 0.11t +.7, where t is the number of ears since 1990. (See Eample.) a. Find (F + M)(t). b. Eplain what (F + M)(t) represents. 0. MODELING WITH MATHEMATICS From 00 to 009, the numbers of cruise ship departures (in thousands) from around the world W and Florida F can be modeled b the equations W(t) =.8t + 17.t + 09.1t + 119 F(t) = 1.t 0.9t + 1.t + 881 where t is the number of ears since 00. a. Find (W F )(t). b. Eplain what (W F )(t) represents. 1. MAKING AN ARGUMENT Your friend claims that the addition of functions and the multiplication of functions are commutative. Is our friend correct? Eplain our reasoning. Section. Performing Function Operations 9
. HOW DO YOU SEE IT? The graphs of the functions f() = 1 and g() = + are shown. Which graph represents the function f + g? the function f g? Eplain our reasoning. A. g B. f. REWRITING A FORMULA For a mammal that weighs w grams, the volume b (in milliliters) of air breathed in and the volume d (in milliliters) of dead space (the portion of the lungs not filled with air) can be modeled b b(w) = 0.007w and d(w) = 0.00w. The breathing rate r (in breaths per minute) of a mammal that weighs w grams can be modeled b 1.1w r (w) = 0.7 b(w) d(w). Simplif r (w) and calculate the breathing rate for bod weights of. grams, 00 grams, and 70,000 grams. 7. PROBLEM SOLVING A mathematician at a lake throws a tennis ball from point A along the water s edge to point B in the water, as shown. His dog, Elvis, first runs along the beach from point A to point D and then swims to fetch the ball at point B.. REASONING The table shows the outputs of the two functions f and g. Use the table to evaluate ( f + g)(), ( f g)(1), ( fg)(), and ( g) f (0). 0 1 f() 0 10 g() 1 1 1 7. THOUGHT PROVOKING Is it possible to write two functions whose sum contains radicals, but whose product does not? Justif our answers.. MATHEMATICAL CONNECTIONS A triangle is inscribed in a square, as shown. Write and simplif a function r in terms of that represents the area of the shaded region. A 0 m B D C 1 m a. Elvis runs at a speed of about. meters per second. Write a function r in terms of that represents the time he spends running from point A to point D. Elvis swims at a speed of about 0.9 meter per second. Write a function s in terms of that represents the time he spends swimming from point D to point B. b. Write a function t in terms of that represents the total time Elvis spends traveling from point A to point D to point B. c. Use a graphing calculator to graph t. Find the value of that minimizes t. Eplain the meaning of this value. Maintaining Mathematical Proficienc Solve the literal equation for n. (Skills Review Handbook) Reviewing what ou learned in previous grades and lessons 8. n 9 = 9. z = 7n + 8nz 0. nb = n z + n 1. = 7b n Determine whether the relation is a function. Eplain. (Skills Review Handbook). (, ), (, ), (1, ), (, 1). ( 1, ), (, 7), (0, ), ( 1, 1). (1, ), (7, ), (, 0), (, 0). (, 8), (, ), (9, ), (, ) 0 Chapter Rational Eponents and Radical Functions
. Inverse of a Function Essential Question How can ou sketch the graph of the inverse of a function? Graphing Functions and Their Inverses CONSTRUCTING VIABLE ARGUMENTS To be proficient in math, ou need to reason inductivel and make a plausible argument. Work with a partner. Each pair of functions are inverses of each other. Use a graphing calculator to graph f and g in the same viewing window. What do ou notice about the graphs? a. f() = + b. f() = + 1 g() = g() = 1 c. f() = g() = +, 0 d. f() = + g() = ( ), Sketching Graphs of Inverse Functions Work with a partner. Use the graph of f to sketch the graph of g, the inverse function of f, on the same set of coordinate aes. Eplain our reasoning. a. b. = 8 8 = = f() 8 8 8 8 = f() 8 8 c. 8 d. 8 = f() = = f() = 8 8 8 8 8 8 Communicate Your Answer. How can ou sketch the graph of the inverse of a function?. In Eploration 1, what do ou notice about the relationship between the equations of f and g? Use our answer to find g, the inverse function of f() =. Use a graph to check our answer. Section. Inverse of a Function 1
. Lesson What You Will Learn Core Vocabular inverse functions, p. Previous input output inverse operations reflection line of reflection Eplore inverses of functions. Find and verif inverses of nonlinear functions. Solve real-life problems using inverse functions. Eploring Inverses of Functions You have used given inputs to find corresponding outputs of = f() for various tpes of functions. You have also used given outputs to find corresponding inputs. Now ou will solve equations of the form = f() for to obtain a general formula for finding the input given a specific output of a function f. Let f() = +. a. Solve = f() for. b. Find the input when the output is 7. Writing a Formula for the Input of a Function Check f( ) = ( ) + = 10 + = 7 a. = + Set equal to f(). = Subtract from each side. = Divide each side b. b. Find the input when = 7. = 7 = 10 Substitute 7 for. Subtract. = Divide. So, the input is when the output is 7. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Solve = f() for. Then find the input(s) when the output is. 1. f() =. f() =. f() = + In Eample 1, notice the steps involved after substituting for in = + and after substituting for in =. = + = Step 1 Multipl b. Step 1 Subtract. Step Add. inverse operations Step Divide b. in the reverse order Chapter Rational Eponents and Radical Functions
UNDERSTANDING MATHEMATICAL TERMS The term inverse functions does not refer to a new tpe of function. Rather, it describes an pair of functions that are inverses. Notice that these steps undo each other. Functions that undo each other are called inverse functions. In Eample 1, ou can use the equation solved for to write the inverse of f b switching the roles of and. f() = + original function g() = inverse function Because inverse functions interchange the input and output values of the original function, the domain and range are also interchanged. Original function: f() = + 1 0 1 f 1 1 7 = Inverse function: g() = g 1 1 7 1 0 1 The graph of an inverse function is a reflection of the graph of the original function. The line of refl ection is =. To find the inverse of a function algebraicall, switch the roles of and, and then solve for. Find the inverse of f() = 1. Finding the Inverse of a Linear Function Method 1 Use inverse operations in the reverse order. Check 9 f The graph of g appears to be a reflection of the graph of f in the line =. g 9 f() = 1 Multipl the input b and then subtract 1. To find the inverse, appl inverse operations in the reverse order. g() = + 1 Add 1 to the input and then divide b. The inverse of f is g() = + 1, or g() = 1 + 1. Method Set equal to f(). Switch the roles of and and solve for. = 1 Set equal to f(). = 1 Switch and. + 1 = Add 1 to each side. + 1 = Divide each side b. The inverse of f is g() = + 1, or g() = 1 + 1. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Find the inverse of the function. Then graph the function and its inverse.. f() =. f() = + 1. f() = 1 Section. Inverse of a Function
Inverses of Nonlinear Functions In the previous eamples, the inverses of the linear functions were also functions. However, inverses are not alwas functions. The graphs of f() = and f() = are shown along with their reflections in the line =. Notice that the inverse of f() = is a function, but the inverse of f() = is not a function. f() = g() = f() = = When the domain of f() = is restricted to onl nonnegative real numbers, the inverse of f is a function. Finding the Inverse of a Quadratic Function Find the inverse of f() =, 0. Then graph the function and its inverse. f() = = Write the original function. Set equal to f(). STUDY TIP If the domain of f were restricted to 0, then the inverse would be g() =. = Switch and. ± = Take square root of each side. The domain of f is restricted to nonnegative values of. So, the range of the inverse must also be restricted to nonnegative values. So, the inverse of f is g() =. f() =, 0 g() = You can use the graph of a function f to determine whether the inverse of f is a function b appling the horizontal line test. Core Concept Horizontal Line Test The inverse of a function f is also a function if and onl if no horizontal line intersects the graph of f more than once. Inverse is a function f Inverse is not a function f Chapter Rational Eponents and Radical Functions
Finding the Inverse of a Cubic Function Consider the function f() = + 1. Determine whether the inverse of f is a function. Then find the inverse. Graph the function f. Notice that no horizontal line intersects the graph more than once. So, the inverse of f is a function. Find the inverse. f() = + 1 Check f g 7 = + 1 Set equal to f(). = + 1 Switch and. 1 = Subtract 1 from each side. 1 = Divide each side b. 1 = Take cube root of each side. So, the inverse of f is g() = 1. Finding the Inverse of a Radical Function Consider the function f() =. Determine whether the inverse of f is a function. Then find the inverse. Graph the function f. Notice that no horizontal line intersects the graph more than once. So, the inverse of f is a function. Find the inverse. = Set equal to f(). = Switch and. 8 f() = Check 9 g = ( ) = ( ) = 1 Square each side. Simplif. Distributive Propert 8 f + 1 = Add 1 to each side. 1 1 1 1 + = Divide each side b. Because the range of f is 0, the domain of the inverse must be restricted to 0. So, the inverse of f is g() = 1 +, where 0. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Find the inverse of the function. Then graph the function and its inverse. 7. f() =, 0 8. f() = + 9. f() = + Section. Inverse of a Function
REASONING ABSTRACTLY Inverse functions undo each other. So, when ou evaluate a function for a specific input, and then evaluate its inverse using the output, ou obtain the original input. Let f and g be inverse functions. If f(a) = b, then g(b) = a. So, in general, f(g()) = and g( f()) =. Verifing Functions Are Inverses Verif that f() = 1 and g() = + 1 are inverse functions. Step 1 Show that f(g()) =. f(g()) = f ( + 1 ) = ( + 1 ) 1 = + 1 1 = Step Show that g( f()) =. g( f()) = g( 1) = 1 + 1 = = Monitoring Progress Determine whether the functions are inverse functions. Help in English and Spanish at BigIdeasMath.com 10. f() = +, g() = 11. f() = 8, g() = Solving Real-Life Problems In man real-life problems, formulas contain meaningful variables, such as the radius r in the formula for the surface area S of a sphere, S = πr. In this situation, switching the variables to find the inverse would create confusion b switching the meanings of S and r. So, when finding the inverse, solve for r without switching the variables. Solving a Multi-Step Problem Find the inverse of the function that represents the surface area of a sphere, S = πr. Then find the radius of a sphere that has a surface area of 100π square feet. The radius r must be positive, so disregard the negative square root. Step 1 Find the inverse of the function. S = πr S π = r S π = r Step Evaluate the inverse when S = 100π. r = 100π π = = The radius of the sphere is feet. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 1. The distance d (in meters) that a dropped object falls in t seconds on Earth is represented b d =.9t. Find the inverse of the function. How long does it take an object to fall 0 meters? Chapter Rational Eponents and Radical Functions
. Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check 1. VOCABULARY In our own words, state the definition of inverse functions.. WRITING Eplain how to determine whether the inverse of a function is also a function.. COMPLETE THE SENTENCE Functions f and g are inverses of each other provided that f(g()) = and g( f()) =.. DIFFERENT WORDS, SAME QUESTION Which is different? Find both answers. Let f() =. Solve = f() for and then switch the roles of and. Write an equation that represents a reflection of the graph of f() = in the -ais. Write an equation that represents a reflection of the graph of f() = in the line =. Find the inverse of f() =. Monitoring Progress and Modeling with Mathematics In Eercises 1, solve = f() for. Then find the input(s) when the output is. (See Eample 1.). f() = +. f() = 7 7. f() = 1 8. f() = + 1 9. f() = 10. f() = 11. f() = ( ) 7. REASONING Determine whether each pair of functions f and g are inverses. Eplain our reasoning. a. 1 0 1 f() 1 7 10 1 7 10 g() 1 0 1 1. f() = ( ) 1 In Eercises 1 0, find the inverse of the function. Then graph the function and its inverse. (See Eample.) 1. f() = 1. f() = 1. f() = + 1. f() = 1 17. f() = + 18. f() = 1 1 19. f() = 1 0. f() = + 1 1. COMPARING METHODS Find the inverse of the function f() = + b switching the roles of and and solving for. Then find the inverse of the function f b using inverse operations in the reverse order. Which method do ou prefer? Eplain. b. c. f() 8 0 g() 8 0 0 f() 10 18 0 g() 1 1 1 1 1 10 18 Section. Inverse of a Function 7
In Eercises 8, find the inverse of the function. Then graph the function and its inverse. (See Eample.). f() =, 0. f() = 9, 0. f() = ( ). f() = ( + ) 7. f() =, 0 8. f() =, 0 ERROR ANALYSIS In Eercises 9 and 0, describe and correct the error in finding the inverse of the function. 9. 0. f () = + = + = + = f () = 1 7, 0 = 1 7 = 1 7 7 = ± 7 = 9. f() = 0. f() = 1. f() = +. f() =. f() = + 1. f() = +. f() = 1. f() = 7 7. WRITING EQUATIONS What is the inverse of the function whose graph is shown? A g() = B g() = + C g() = D g() = + 1 8. WRITING EQUATIONS What is the inverse of f() = 1? A g() = B g() = C g() = D g() = USING TOOLS In Eercises 1, use the graph to determine whether the inverse of f is a function. Eplain our reasoning. 1.. 10 1 10 f f.. 10 8 8 f f 8 In Eercises 9, determine whether the functions are inverse functions. (See Eample.) 9. f() = 9, g() = + 9 0. f() =, g() = + 1. f() = + 9, g() = 9. f() = 7 /, g() = ( + 7 ) /. MODELING WITH MATHEMATICS The maimum hull speed v (in knots) of a boat with a displacement hull can be approimated b v = 1., where is the waterline length (in feet) of the boat. Find the inverse function. What waterline length is needed to achieve a maimum speed of 7. knots? (See Eample 7.) In Eercises, determine whether the inverse of f is a function. Then find the inverse. (See Eamples and.). f() = 1. f() = + Waterline length 7. f() = + 8. f() = 8 Chapter Rational Eponents and Radical Functions
. MODELING WITH MATHEMATICS Elastic bands can be used for eercising to provide a range of resistance. The resistance R (in pounds) of a band can be modeled b R = L, where L is the total 8 length (in inches) of the stretched band. Find the inverse function. What length of the stretched band provides 19 pounds of resistance? unstretched stretched ANALYZING RELATIONSHIPS In Eercises 8, match the graph of the function with the graph of its inverse... 7. 8. A. B. C. D. 9. REASONING You and a friend are plaing a numberguessing game. You ask our friend to think of a positive number, square the number, multipl the result b, and then add. Your friend s final answer is. What was the original number chosen? Justif our answer. 0. MAKING AN ARGUMENT Your friend claims that ever quadratic function whose domain is restricted to nonnegative values has an inverse function. Is our friend correct? Eplain our reasoning. 1. PROBLEM SOLVING When calibrating a spring scale, ou need to know how far the spring stretches for various weights. Hooke s Law states that the length a spring stretches is proportional to the weight attached to it. A model for one scale is = 0.w +, where is the total length (in inches) of the stretched spring and w is the weight (in pounds) of the object. a. Find the inverse function. Describe what it represents. unweighted spring 0.w b. You place a melon on the scale, and the spring stretches to a total length of. inches. Determine the weight of the melon. c. Verif that the function = 0.w + and the inverse model in part (a) are inverse functions.. THOUGHT PROVOKING Do functions of the form = m/n, where m and n are positive integers, have inverse functions? Justif our answer with eamples.. PROBLEM SOLVING At the start of a dog sled race in Anchorage, Alaska, the temperature was C. B the end of the race, the temperature was 10 C. The formula for converting temperatures from degrees Fahrenheit F to degrees Celsius C is C = (F ). 9 a. Find the inverse function. Describe what it represents. b. Find the Fahrenheit temperatures at the start and end of the race. c. Use a graphing calculator to graph the original function and its inverse. Find the temperature that is the same on both temperature scales. Not drawn to scale spring with weight attached Section. Inverse of a Function 9
. PROBLEM SOLVING The surface area A (in square meters) of a person with a mass of 0 kilograms can be approimated b A = 0.19h 0.9, where h is the height (in centimeters) of the person. a. Find the inverse function. Then estimate the height of a 0-kilogram person who has a bod surface area of 1. square meters. b. Verif that function A and the inverse model in part (a) are inverse functions. USING STRUCTURE In Eercises 8, match the function with the graph of its inverse.. f() =. f() = + 7. f() = + 1 9. DRAWING CONCLUSIONS Determine whether the statement is true or false. Eplain our reasoning. a. If f() = n and n is a positive even integer, then the inverse of f is a function. b. If f() = n and n is a positive odd integer, then the inverse of f is a function. 70. HOW DO YOU SEE IT? The graph of the function f is shown. Name three points that lie on the graph of the inverse of f. Eplain our reasoning. f 8. f() = 1 + A. C. 8 B. D. 8 Maintaining Mathematical Proficienc 71. ABSTRACT REASONING Show that the inverse of an linear function f() = m + b, where m 0, is also a linear function. Identif the slope and -intercept of the graph of the inverse function in terms of m and b. 7. CRITICAL THINKING Consider the function f() =. a. Graph f() = and eplain wh it is its own inverse. Also, verif that f() = is its own inverse algebraicall. b. Graph other linear functions that are their own inverses. Write equations of the lines ou graphed. c. Use our results from part (b) to write a general equation describing the famil of linear functions that are their own inverses. Simplif the epression. Write our answer using onl positive eponents. (Skills Review Handbook) 7. ( ) 7. 7. ) 7. ( Describe the -values for which the function is increasing, decreasing, positive, and negative. (Section.1) 77. 78. Reviewing what ou learned in previous grades and lessons 79. 1 1 = 1 = 1 = + 1 1 0 Chapter Rational Eponents and Radical Functions
.. What Did You Learn? Core Vocabular radical equation, p. 18 etraneous solutions, p. 19 inverse functions, p. Core Concepts Section. Solving Radical Equations, p. 18 Solving Radical Inequalities, p. 1 Section. Operations on Functions, p. Section. Eploring Inverses of Functions, p. Inverses of Nonlinear Functions, p. Horizontal Line Test, p. Mathematical Practices 1. How did ou find the endpoints of the range in part (b) of Eercise on page?. How did ou use structure in Eercise 7 on page?. How can ou evaluate the reasonableness of the results in Eercise 7 on page 0?. How can ou use a graphing calculator to check our answers in Eercises 9 on page 8? Performance Task: The Heartbeat Hpothesis Biologists use mathematical functions to model characteristics of different species, such as heart rate, bod mass, and life span. How can the functions be combined to tell us even more? To eplore the answer to this question and more, check out the Performance Task and Real-Life STEM video at BigIdeasMath.com. 1
Chapter Review.1 nth Roots and Rational Eponents (pp. 19 198) Dnamic Solutions available at BigIdeasMath.com a. Evaluate 8 / without using a calculator. Rational Eponent Form Radical Form 8 / = (8 1/ ) = = 1 8 / = ( 8 ) = = 1 b. Find the real solution(s) of = 80. = 80 = = ± = or = Write original equation. Add to each side. Take fourth root of each side. Simplif. The solutions are = and =. Evaluate the epression without using a calculator. 1. 8 7/. 9 /. ( 7) / Find the real solution(s) of the equation. Round our answer to two decimal places when appropriate.. + 17 =. 7 = 189. ( + 8) = 1. Properties of Rational Eponents and Radicals (pp. 199 0) a. Use the properties of rational eponents to simplif ( 1/ ). ( 1/ 1/ ) 1/ = [ ( 1/ ) ] = (7 1/ ) = = 81 b. Write 1 1 8 z 7 in simplest form. Assume all variables are positive. 1 1 8 z 7 = 1 1 8 z z Simplif the epression. 7. ( 1/ ) / Factor out perfect fourth powers. = 1 1 8 z z Product Propert of Radicals = z z Simplif. 8. 8 9. 1 9 10. 8 + 8 11. 8 1. ( / / ) 1/ Simplif the epression. Assume all variables are positive. 1. 1z 9 1. 1/ z / z 1. 10z z 0z Chapter Rational Eponents and Radical Functions
. Graphing Radical Functions (pp. 07 1) Describe the transformation of f() = represented b g() = +. Then graph each function. g Notice that the function is of the form g() = a h, where a = and h =. So, the graph of g is a vertical stretch b a factor of and a translation units left of the graph of f. f Describe the transformation of f represented b g. Then graph each function. 1. f() =, g() = 17. f() =, g() = 18. Let the graph of g be a reflection in the -ais, followed b a translation 7 units to the right of the graph of f() =. Write a rule for g. 19. Use a graphing calculator to graph = 8. Identif the verte and the direction that the parabola opens. 0. Use a graphing calculator to graph + = 81. Identif the center, radius, and the intercepts.. Solving Radical Equations and Inequalities (pp. 17 ) Solve + < 18. Step 1 Solve for. Step + < 18 Write the original inequalit. + < Divide each side b. + < 9 Square each side. < 7 Subtract from each side. Consider the radicand. + 0 The radicand cannot be negative. Subtract from each side. Check 0 = 18 = + Intersection 1 X=7 Y=18 9 So, the solution is < 7. Solve the equation. Check our solution(s). 1. + 1 = 0. = 1 1. () / = Solve the inequalit.. + > 17. 8 <. 7 1 7. In a tsunami, the wave speeds (in meters per second) can be modeled b s(d ) = 9.8d, where d is the depth (in meters) of the water. Estimate the depth of the water when the wave speed is 00 meters per second. Chapter Chapter Review
. Performing Function Operations (pp. 0) g) Let f() = / and g() = 1/. Find ( f () and state the domain. Then evaluate the quotient when = 81. ( f f() () = g() = / 1/ = (/ 1/) = / g) The functions f and g each have the same domain: all nonnegative real numbers. Because g(0) = 0, the domain of f is restricted to all positive real numbers. g When = 81, the value of the quotient is ( f g) (81) = (81)/ = (81 1/ ) = () = () = 8. 8. Let f() = and g() =. Find ( fg)() and ( f () and state the domain of each. Then evaluate ( fg)() and ( g) f (). 9. Let f() = + 1 and g() = +. Find ( f + g)() and ( f g)() and state the domain of each. Then evaluate ( f + g)( ) and ( f g)( ). g). Inverse of a Function (pp. 1 0) Consider the function f() = ( + ). Determine whether the inverse of f is a function. Then find the inverse. f() = ( + ) Graph the function f. Notice that no horizontal line intersects the graph more than once. So, the inverse of f is a function. Find the inverse. = ( + ) Set equal to f(). = ( + ) Switch and. = + Take cube root of each side. = So, the inverse of f is g() =. Subtract from each side. Check f 8 Find the inverse of the function. Then graph the function and its inverse. 1 0. f() = + 10 1. f() = + 8, 0. f() = 9. f() = + 1 8 g 1 Determine whether the functions are inverse functions.. f() = ( 11), g() = 1 ( + 11). f() = +, g() = +. On a certain da, the function that gives U.S. dollars in terms of British pounds is d = 1.87p, where d represents U.S. dollars and p represents British pounds. Find the inverse function. Then find the number of British pounds equivalent to 100 U.S. dollars. Chapter Rational Eponents and Radical Functions 1
Chapter Test 1. Solve the inequalit 1 and the equation = 1. Describe the similarities and differences in solving radical equations and radical inequalities. Describe the transformation of f represented b g. Then write a rule for g.. f() =. f() =. f() = (, 1) (, 0) g g ( 1, ) (0, 0) (1, ) ( 1, 0) (1, ) (0, ) g Simplif the epression. Eplain our reasoning.. /. ( 7) / 7. 8 11 z 8. 9. Write two functions whose graphs are translations of the graph of =. The first function should have a domain of. The second function should have a range of. 10. In bowling, a handicap is a change in score to adjust for differences in the abilities of plaers. You belong to a bowling league in which our handicap h is determined using the formula h = 0.9(00 a), where a is our average score. Find the inverse of the model. Then find the average score for a bowler whose handicap is. 11. The basal metabolic rate of an animal is a measure of the amount of calories burned at rest for basic functioning. Kleiber s law states that an animal s basal metabolic rate R (in kilocalories per da) can be modeled b R = 7.w /, where w is the mass (in kilograms) of the animal. Find the basal metabolic rates of each animal in the table. 1. Let f() = / and g() = /. Find ( f + g)() and ( f g)() and state the domain of each. Then evaluate ( f + g)() and ( f g)(). 1. Let f() = 1 / and g() = 8. Find ( fg)() and ( f of each. Then evaluate ( fg)(1) and ( g) f (1). g) () and state the domain 1. A football plaer jumps to catch a pass. The maimum height h (in feet) of the plaer above the ground is given b the function h = 1 s, where s is the initial speed (in feet per second) of the plaer. Find the inverse of the function. Use the inverse to find the initial speed of the plaer shown. Verif that the functions are inverse functions. Animal Mass (kilograms) rabbit. sheep 0 human 70 lion 10 ft Chapter Chapter Test
Cumulative Assessment 1. Identif three pairs of equivalent epressions. Assume all variables are positive. Justif our answer. a a 1/n n a n a 1/n ( a ) n a n n a a n. The graph represents the function f() = ( ) +. Choose the correct values to complete the function. f 1 1. In rowing, the boat speed s (in meters per second) can be modeled b s =. 9 n, where n is the number of rowers. a. Find the boat speeds for crews of people, people, and 8 people. b. Does the boat speed double when the number of rowers doubles? Eplain. c. Find the time (in minutes) it takes each crew in part (a) to complete a 000-meter race.. A polnomial function fits the data in the table. Use finite differences to find the degree of the function and complete the table. Eplain our reasoning. 1 0 1 f() 8 8 18. A cit charges $.70 per 100 cubic feet of water used. a. What is the cost of filling the pool to a depth of feet? b. There are about 7.8 gallons in 1 cubic foot of water. Epress the cost of water in dollars per 100 gallons. c. You add water to the -foot-deep pool in part (a) until the water is 1 meter deep. The densit of water is about 1000 kilograms per cubic meter. What is the mass of the additional water? 1 ft ft Chapter Rational Eponents and Radical Functions