Inverse Functions - Beau Chene High School Home Page

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EXAMPLE 5 Finding an Inverse Function Consider the function ƒ(x) = 1 2 x3 º 2. Determine whether the inverse of ƒ is a function. Then find the inverse. SOLUTION Begin by graphing the function and noticing that no horizontal line intersects the graph more than once. This tells you that the inverse of ƒ is itself a function. To find an equation for ƒ º1 , complete the following steps. 1 y 1 x ƒ(x) = 1 2 x3 º 2 Write original function. y = 1 2 x3 º 2 Replace ƒ(x) with y. x = 1 2 y3 º 2 Switch x and y. x + 2 = 1 2 y3 Add 2 to each side. 2x + 4 = y 3 Multiply each side by 2. 3 2x+ 4 = y Take cube root of each side. The inverse function is ƒ º1 (x) = 3 2x+ 4. EXAMPLE 6 Writing an Inverse Model FOCUS ON APPLICATIONS ASTRONOMY Near the end of a star’s life the star will eject gas, forming a planetary nebula. The Ring Nebula is an example of a planetary nebula. The volume V (in cubic kilometers) of this nebula can be modeled by V = (9.01 ª 10 26 )t 3 where t is the age (in years) of the nebula. Write the inverse model that gives the age of the nebula as a function of its volume. Then determine the approximate age of the Ring Nebula given that its volume is about 1.5 ª 10 38 cubic kilometers. SOLUTION 3 V 9.01 ª 10 26 V = t 9.01 ª 10 26 V = (9.01 ª 10 26 )t 3 Write original model. = t 3 Isolate power. Take cube root of each side. ASTRONOMY The Ring Nebula is part of the constellation Lyra. The radius of the nebula is expanding at an average rate of about 5.99 10 8 kilometers per year. APPLICATION LINK www.mcdougallittell.com REAL INTERNET LIFE (1.04 ª 10 º9 ) 3 V = t Simplify. To find the age of the nebula, substitute 1.5 ª 10 38 for V. t = (1.04 ª 10 º9 ) 3 V Write inverse model. = (1.04 ª 10 º9 ) 3 1.5 ª 10 38 Substitute for V. ≈ 5500 Use a calculator. The Ring Nebula is about 5500 years old. 7.4 Inverse Functions 425
GUIDED PRACTICE Vocabulary Check ✓ Concept Check ✓ Skill Check ✓ 1. Explain how to use the horizontal line test to determine if an inverse relation is an inverse function. 2. Describe how the graph of a relation and the graph of its inverse are related. 3. Explain the steps in finding an equation for an inverse function. Find the inverse relation. 4. x 1 2 3 4 5 5. y º1 º2 º3 º4 º5 x º4 º2 0 2 4 y 2 1 0 1 2 Find an equation for the inverse relation. 6. y = 5x 7. y = 2x º 1 8. y = º 2 3 x + 6 Verify that ƒ and g are inverse functions. /3 9. ƒ(x) = 8x 3 , g(x) = x1 2 10. ƒ(x) = 6x + 3, g(x) = 1 6 x º 1 2 Find the inverse function. PRACTICE AND APPLICATIONS 11. ƒ(x) = 3x 4 , x ≥ 0 12. ƒ(x) = 2x 3 + 1 13. The graph of ƒ(x) = º|x| + 1 is shown. Is the inverse of ƒ a function? Explain. 1 y Ex. 13 3 x STUDENT HELP Extra Practice to help you master skills is on p. 949. INVERSE RELATIONS Find the inverse relation. 14. x 1 4 1 0 1 15. x 1 º2 4 2 º2 y 3 º1 6 º3 9 y 0 3 º2 2 º1 FINDING INVERSES Find an equation for the inverse relation. 16. y = º2x + 5 17. y = 3x º 3 18. y = 1 2 x + 6 19. y = º 4 x + 11 20. y = 11x º 5 21. y = º12x + 7 5 22. y = 3x º 1 4 23. y = 8x º 13 24. y = º3 7 x + 5 7 STUDENT HELP HOMEWORK HELP Example 1: Exs. 14–24 Example 2: Exs. 25–32 Example 3: Exs. 57–59 Example 4: Exs. 33–41 Example 5: Exs. 42–56 Example 6: Exs. 60–62 VERIFYING INVERSES Verify that ƒ and g are inverse functions. 25. ƒ(x) = x + 7, g(x) = x º 7 26. ƒ(x) = 3x º 1, g(x) = 1 3 x + 1 3 27. ƒ(x) = 1 2 x + 1, g(x) = 2x º 2 28. ƒ(x) = º2x + 4, g(x) = º1 2 x + 2 29. ƒ(x) = 3x 3 + 1, g(x) = x 1 3 1/3 30. ƒ(x) = 1 3 x2 , x ≥ 0; g(x) = (3x) 1/2 x 31. ƒ(x) = +2 , g(x) = 5 7xº 2 7 32. ƒ(x) = 256x 4 , x ≥ 0; g(x) = 4 x 4 426 Chapter 7 Powers, Roots, and Radicals
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