SECTION 5.3 Exponential Functions 307 9. Interactive Figure Exercise Exploring Exponential Functions Open the “Exponential Functions” interactive figure, which is available in the Video & Resource Library of MyLab Math (under Sullivan Interactive Figures). (a) Use the sliders to set the value of c to 1, a to 2, h to 0, and k to 0. Now, use the slider to increase the value of h from 0 to 4.The graph shifts (horizontally/ vertically) h units (left/right/up/down). (b) Use the sliders to set the value of c to 1, a to 2, h to 0, and k to 0. Now, use the slider to decrease the value of h from 0 to −4. (i) What is the equation of the function when = − h 3? (ii) As h decreases from 0 to −4, the graph shifts (horizontally/vertically) h units (left/right/up/down). (c) Use the sliders to set the value of c to 1, a to 2, h to 0, and k to 0. Now, use the slider to increase the value of k from 0 to 4. The graph shifts (horizontally/ vertically) h units (left/right/up/down). (d) Use the sliders to set the value of c to 1, a to 2, h to 0, and k to 0. Now, use the slider to decrease the value of k from 0 to −4. (i) What is the equation of the function when = − k 3? (ii) What is the horizontal asymptote when = − k 3? (iii) As k decreases from 0 to −4, the graph shifts (horizontally/vertically) k units (left/right/up/down). (e) Use the sliders to set the value of c to 1, a to 2, h to 0, and k to 0. Now, use the slider to increase the value of c from 1 to 4. The graph is vertically (stretched/ compressed) by a factor of c. When = c 2, the points 1, , 0, , ( ) ( ) − and 1, ( ) are on the graph of f. (f) In the graph of ( ) = ⋅ f x c 2x with < c 0, the graph is (increasing/decreasing) over its domain. 10. A(n) is a function of the form ( ) = > ≠ f x Ca a a , where 0, 1, x and ≠ C 0 are real numbers. The base a is the and C is the . 11. For an exponential function ( ) ( ) = + = f x Ca f x f x , ( 1) x . 12. True or False The domain of the function ( ) = > ≠ f x a a a , where 0 and 1, x is the set of all real numbers. 13. True or False The function ( ) = f x ex is increasing and is one-to-one. 14. The graph of every exponential function ( ) = f x a ,x where > a 0 and ≠ a 1, contains the three points: 1, , 0, , ( ) ( ) − and 1, . ( ) 15. If = 3 3 , x 4 then = x . 16. True or False The graphs of = y 3x and ( ) = y 1 3 x are identical. 17. Multiple Choice Which exponential function is increasing? (a) ( ) = f x 0.5x (b) ( ) ( ) = f x 5 2 x (c) ( ) ( ) = f x 2 3 x (d) ( ) = f x 0.9x 18. Multiple Choice The range of the function ( ) = f x a ,x where > a 0 and ≠ a 1, is the interval (a) ( ) −∞ ∞, (b) ( ) −∞, 0 (c) ( )∞ 0, (d) [ )∞ 0, Skill Building In Problems 19–30, approximate each number using a calculator. Express your answer rounded to three decimal places. 19. (a) 23.14 (b) 23.141 (c) 23.1415 (d) π2 27. e1.2 28. −e 1.3 20. (a) 22.7 (b) 22.71 (c) 22.718 (d) 2e 21. (a) 3.12.7 (b) 3.142.71 (c) 3.1412.718 (d) πe 22. (a) 2.73.1 (b) 2.713.14 (c) 2.7183.141 (d) πe 23. ( ) +1 0.04 6 24. ( ) +1 0.09 12 24 25. ( ) 8.4 1 3 2.9 26. ( ) 158 5 6 8.63 29. ⋅ e 125 0.026 7 30. − ⋅ e 83.6 0.157 9.5 In Problems 31–38, determine whether the given function is linear, exponential, or neither. For those that are linear functions, find a linear function that models the data; for those that are exponential, find an exponential function that models the data. x f x( ) −1 3 0 6 1 12 2 18 3 30 x g x( ) −1 2 0 5 1 8 2 11 3 14 31. 32. x H x( ) −1 5 4 0 5 1 20 2 80 3 320 33. x F x( ) −1 2 3 0 1 1 3 2 2 9 4 3 27 8 34.
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