Inverse of a Function

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1 . Inverse o a Function Essential Question How can ou sketch the graph o the inverse o a unction? Graphing Functions and Their Inverses CONSTRUCTING VIABLE ARGUMENTS To be proicient in math, ou need to reason inductivel and make a plausible argument. Work with a partner. Each pair o unctions are inverses o each other. Use a graphing calculator to graph and g in the same viewing window. What do ou notice about the graphs? a. () = + b. () = + g() = g() = c. () = g() = +, 0 d. () = + g() = ( ), Sketching Graphs o Inverse Functions Work with a partner. Use the graph o to sketch the graph o g, the inverse unction o, on the same set o coordinate aes. Eplain our reasoning. a. b. = = = () = () c. d. = () = = () = Communicate Your Answer. How can ou sketch the graph o the inverse o a unction?. In Eploration, what do ou notice about the relationship between the equations o and g? Use our answer to ind g, the inverse unction o () =. Use a graph to check our answer. Section. Inverse o a Function

2 . Lesson What You Will Learn Core Vocabular inverse unctions, p. Previous input output inverse operations relection line o relection Eplore inverses o unctions. Find and veri inverses o nonlinear unctions. Solve real-lie problems using inverse unctions. Eploring Inverses o Functions You have used given inputs to ind corresponding outputs o = () or various tpes o unctions. You have also used given outputs to ind corresponding inputs. Now ou will solve equations o the orm = () or to obtain a general ormula or inding the input given a speciic output o a unction. Let () = +. a. Solve = () or. b. Find the input when the output is 7. Writing a Formula or the Input o a Function Check ( 5) = ( 5) + = 0 + = 7 a. = + Set equal to (). = Subtract rom each side. = Divide each side b. b. Find the input when = 7. = 7 Substitute 7 or. = 0 Subtract. = 5 Divide. So, the input is 5 when the output is 7. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Solve = () or. Then ind the input(s) when the output is.. () =. () =. () = + In Eample, notice the steps involved ater substituting or in = + and ater substituting or in =. = + = Step Multipl b. Step Subtract. Step Add. inverse operations Step Divide b. in the reverse order Chapter Rational Eponents and Radical Functions

3 UNDERSTANDING MATHEMATICAL TERMS The term inverse unctions does not reer to a new tpe o unction. Rather, it describes an pair o unctions that are inverses. Notice that these steps undo each other. Functions that undo each other are called inverse unctions. In Eample, ou can use the equation solved or to write the inverse o b switching the roles o and. () = + original unction g() = inverse unction Because inverse unctions interchange the input and output values o the original unction, the domain and range are also interchanged. Original unction: () = Inverse unction: g() = = g The graph o an inverse unction is a rel ection o the graph o the original unction. The line o rel ection is =. To ind the inverse o a unction algebraicall, switch the roles o and, and then solve or. Find the inverse o () =. Finding the Inverse o a Linear Function Method Use inverse operations in the reverse order. Check 9 The graph o g appears to be a relection o the graph o in the line =. g 9 () = Multipl the input b and then subtract. To ind the inverse, appl inverse operations in the reverse order. g() = + Add to the input and then divide b. The inverse o is g() = +, or g() = +. Method Set equal to (). Switch the roles o and and solve or. = Set equal to (). = Switch and. + = Add to each side. + = Divide each side b. The inverse o is g() = +, or g() = +. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Find the inverse o the unction. Then graph the unction and its inverse.. () = 5. () = +. () = Section. Inverse o a Function

4 Inverses o Nonlinear Functions In the previous eamples, the inverses o the linear unctions were also unctions. However, inverses are not alwas unctions. The graphs o () = and () = are shown along with their relections in the line =. Notice that the inverse o () = is a unction, but the inverse o () = is not a unction. () = g() = () = = When the domain o () = is restricted to onl nonnegative real numbers, the inverse o is a unction. Finding the Inverse o a Quadratic Function Find the inverse o () =, 0. Then graph the unction and its inverse. () = = Write the original unction. Set equal to (). STUDY TIP I the domain o were restricted to 0, then the inverse would be g() =. = Switch and. ± = Take square root o each side. The domain o is restricted to nonnegative values o. So, the range o the inverse must also be restricted to nonnegative values. So, the inverse o is g() =. () =, 0 g() = You can use the graph o a unction to determine whether the inverse o is a unction b appling the horizontal line test. Core Concept Horizontal Line Test The inverse o a unction is also a unction i and onl i no horizontal line intersects the graph o more than once. Inverse is a unction Inverse is not a unction Chapter Rational Eponents and Radical Functions

5 Finding the Inverse o a Cubic Function Consider the unction () = +. Determine whether the inverse o is a unction. Then ind the inverse. Graph the unction. Notice that no horizontal line intersects the graph more than once. So, the inverse o is a unction. Find the inverse. () = + Check 5 5 g 7 = + Set equal to (). = + Switch and. = Subtract rom each side. = Divide each side b. = Take cube root o each side. So, the inverse o is g() =. Finding the Inverse o a Radical Function Consider the unction () =. Determine whether the inverse o is a unction. Then ind the inverse. Graph the unction. Notice that no horizontal line intersects the graph more than once. So, the inverse o is a unction. Find the inverse. = Set equal to (). = Switch and. () = Check 9 g = ( ) = ( ) = Square each side. Simpli. Distributive Propert + = Add to each side. + = Divide each side b. Because the range o is 0, the domain o the inverse must be restricted to 0. So, the inverse o is g() = +, where 0. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Find the inverse o the unction. Then graph the unction and its inverse. 7. () =, 0. () = + 9. () = + Section. Inverse o a Function 5

6 REASONING ABSTRACTLY Inverse unctions undo each other. So, when ou evaluate a unction or a speciic input, and then evaluate its inverse using the output, ou obtain the original input. Let and g be inverse unctions. I (a) = b, then g(b) = a. So, in general, (g()) = and g( ()) =. Veriing Functions Are Inverses Veri that () = and g() = + are inverse unctions. Step Show that (g()) =. (g()) = ( + ) = ( + ) = + = Step Show that g( ()) =. g( ()) = g( ) = + = = Monitoring Progress Determine whether the unctions are inverse unctions. Help in English and Spanish at BigIdeasMath.com 0. () = + 5, g() = 5. () =, g() = Solving Real-Lie Problems In man real-lie problems, ormulas contain meaningul variables, such as the radius r in the ormula or the surace area S o a sphere, S = πr. In this situation, switching the variables to ind the inverse would create conusion b switching the meanings o S and r. So, when inding the inverse, solve or r without switching the variables. Solving a Multi-Step Problem Find the inverse o the unction that represents the surace area o a sphere, S = πr. Then ind the radius o a sphere that has a surace area o 00π square eet. The radius r must be positive, so disregard the negative square root. Step Find the inverse o the unction. S = πr S π = r S π = r Step Evaluate the inverse when S = 00π. r = 00π π = 5 = 5 The radius o the sphere is 5 eet. Monitoring Progress Help in English and Spanish at BigIdeasMath.com. The distance d (in meters) that a dropped object alls in t seconds on Earth is represented b d =.9t. Find the inverse o the unction. How long does it take an object to all 50 meters? Chapter Rational Eponents and Radical Functions

7 . Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. VOCABULARY In our own words, state the deinition o inverse unctions.. WRITING Eplain how to determine whether the inverse o a unction is also a unction.. COMPLETE THE SENTENCE Functions and g are inverses o each other provided that (g()) = and g( ()) =.. DIFFERENT WORDS, SAME QUESTION Which is dierent? Find both answers. Let () = 5. Solve = () or and then switch the roles o and. Write an equation that represents a relection o the graph o () = 5 in the -ais. Write an equation that represents a relection o the graph o () = 5 in the line =. Find the inverse o () = 5. Monitoring Progress and Modeling with Mathematics In Eercises 5, solve = () or. Then ind the input(s) when the output is. (See Eample.) 5. () = + 5. () = 7 7. () =. () = + 9. () = 0. () = 5. () = ( ) 7. REASONING Determine whether each pair o unctions and g are inverses. Eplain our reasoning. a. 0 () g() 0. () = ( 5) In Eercises 0, ind the inverse o the unction. Then graph the unction and its inverse. (See Eample.). () =. () = 5. () = + 5. () = 7. () = +. () = 9. () = 0. () = COMPARING METHODS Find the inverse o the unction () = + b switching the roles o and and solving or. Then ind the inverse o the unction b using inverse operations in the reverse order. Which method do ou preer? Eplain. b. c. 5 () 0 5 g() 0 0 () 0 0 g() 0 Section. Inverse o a Function 7

8 In Eercises, ind the inverse o the unction. Then graph the unction and its inverse. (See Eample.). () =, 0. () = 9, 0 5. () = ( ). () = ( + ) 7. () =, 0. () =, 0 ERROR ANALYSIS In Eercises 9 and 0, describe and correct the error in inding the inverse o the unction () = + = + = + = () = 7, 0 = 7 = 7 7 = ± 7 = 9. () = 5 0. () = 5. () = +. () = 5. () = +. () = + 5. () = 5. () = 7 7. WRITING EQUATIONS What is the inverse o the unction whose graph is shown? A g() = B g() = + C g() = D g() = +. WRITING EQUATIONS What is the inverse o () =? A g() = B g() = C g() = D g() = USING TOOLS In Eercises, use the graph to determine whether the inverse o is a unction. Eplain our reasoning In Eercises 9 5, determine whether the unctions are inverse unctions. (See Eample.) 9. () = 9, g() = () =, g() = + 5. () = , g() = () = 7 /, g() = ( + 7 ) / 5. MODELING WITH MATHEMATICS The maimum hull speed v (in knots) o a boat with a displacement hull can be approimated b v =., where is the waterline length (in eet) o the boat. Find the inverse unction. What waterline length is needed to achieve a maimum speed o 7.5 knots? (See Eample 7.) In Eercises 5, determine whether the inverse o is a unction. Then ind the inverse. (See Eamples and 5.) 5. () =. () = + Waterline length 7. () = +. () = Chapter Rational Eponents and Radical Functions

9 5. MODELING WITH MATHEMATICS Elastic bands can be used or eercising to provide a range o resistance. The resistance R (in pounds) o a band can be modeled b R = L 5, where L is the total length (in inches) o the stretched band. Find the inverse unction. What length o the stretched band provides 9 pounds o resistance? unstretched stretched ANALYZING RELATIONSHIPS In Eercises 55 5, match the graph o the unction with the graph o its inverse A. B. C. D. 59. REASONING You and a riend are plaing a numberguessing game. You ask our riend to think o a positive number, square the number, multipl the result b, and then add. Your riend s inal answer is 5. What was the original number chosen? Justi our answer. 0. MAKING AN ARGUMENT Your riend claims that ever quadratic unction whose domain is restricted to nonnegative values has an inverse unction. Is our riend correct? Eplain our reasoning.. PROBLEM SOLVING When calibrating a spring scale, ou need to know how ar the spring stretches or various weights. Hooke s Law states that the length a spring stretches is proportional to the weight attached to it. A model or one scale is = 0.5w +, where is the total length (in inches) o the stretched spring and w is the weight (in pounds) o the object. a. Find the inverse unction. Describe Not drawn to scale what it represents. b. You place a melon on the scale, and the spring stretches to a total length o 5.5 inches. Determine the weight o the melon. c. Veri that the unction = 0.5w + and the inverse model in part (a) are inverse unctions.. THOUGHT PROVOKING Do unctions o the orm = m/n, where m and n are positive integers, have inverse unctions? Justi our answer with eamples.. PROBLEM SOLVING At the start o a dog sled race in Anchorage, Alaska, the temperature was 5 C. B the end o the race, the temperature was 0 C. The ormula or converting temperatures rom degrees Fahrenheit F to degrees Celsius C is C = 5 (F ). 9 a. Find the inverse unction. Describe what it represents. unweighted spring 0.5w b. Find the Fahrenheit temperatures at the start and end o the race. c. Use a graphing calculator to graph the original unction and its inverse. Find the temperature that is the same on both temperature scales. spring with weight attached Section. Inverse o a Function 9

10 . PROBLEM SOLVING The surace area A (in square meters) o a person with a mass o 0 kilograms can be approimated b A = 0.95h 0.9, where h is the height (in centimeters) o the person. a. Find the inverse unction. Then estimate the height o a 0-kilogram person who has a bod surace area o. square meters. b. Veri that unction A and the inverse model in part (a) are inverse unctions. USING STRUCTURE In Eercises 5, match the unction with the graph o its inverse. 5. () =. () = + 7. () = + 9. DRAWING CONCLUSIONS Determine whether the statement is true or alse. Eplain our reasoning. a. I () = n and n is a positive even integer, then the inverse o is a unction. b. I () = n and n is a positive odd integer, then the inverse o is a unction. 70. HOW DO YOU SEE IT? The graph o the unction is shown. Name three points that lie on the graph o the inverse o. Eplain our reasoning.. () = + A. C. B. D. Maintaining Mathematical Proicienc 7. ABSTRACT REASONING Show that the inverse o an linear unction () = m + b, where m 0, is also a linear unction. Identi the slope and -intercept o the graph o the inverse unction in terms o m and b. 7. CRITICAL THINKING Consider the unction () =. a. Graph () = and eplain wh it is its own inverse. Also, veri that () = is its own inverse algebraicall. b. Graph other linear unctions that are their own inverses. Write equations o the lines ou graphed. c. Use our results rom part (b) to write a general equation describing the amil o linear unctions that are their own inverses. Simpli the epression. Write our answer using onl positive eponents. (Skills Review Handbook) 7. ( ) ) 5 7. ( Describe the -values or which the unction is increasing, decreasing, positive, and negative. (Section.) Reviewing what ou learned in previous grades and lessons 79. = = = Chapter Rational Eponents and Radical Functions

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