Precalculus An Investigation of Functions

Transcription

1 Precalculus An Investigation of Functions David Lippman Melonie Rasmussen Edition 1. This book is also available to read free online at If you want a printed copy, buying from the bookstore is cheaper than printing yourself.

2 ii Copyright 01 David Lippman and Melonie Rasmussen This text is licensed under a Creative Commons Attribution-Share Alike.0 United States License. To view a copy of this license, visit or send a letter to Creative Commons, 171 Second Street, Suite 00, San Francisco, California, 94105, USA. You are free: to Share to copy, distribute, display, and perform the work to Remix to make derivative works Under the following conditions: Attribution. You must attribute the work in the manner specified by the author or licensor (but not in any way that suggests that they endorse you or your use of the work). Share Alike. If you alter, transform, or build upon this work, you may distribute the resulting work only under the same, similar or a compatible license. With the understanding that: Waiver. Any of the above conditions can be waived if you get permission from the copyright holder. Other Rights. In no way are any of the following rights affected by the license: Your fair dealing or fair use rights; Apart from the remix rights granted under this license, the author's moral rights; Rights other persons may have either in the work itself or in how the work is used, such as publicity or privacy rights. Notice For any reuse or distribution, you must make clear to others the license terms of this work. The best way to do this is with a link to this web page: In addition to these rights, we give explicit permission to remix small portions of this book (less than 10% cumulative) into works that are CC-BY, CC-BY-SA-NC, or GFDL licensed. Selected exercises were remixed from Precalculus by D.H. Collingwood and K.D. Prince, originally licensed under the GNU Free Document License, with permission from the authors. Cover Photo by David Lippman, of artwork by John Rogers Lituus, 010 Dichromatic glass and aluminum Washington State Arts Commission in partnership with Pierce College This is the fourth official version of Edition 1. It contains typo corrections and language clarification, but is page number and problem set number equivalent to the original Edition 1.

3 1 Chapter Section.5 Exercises For each function, find a domain on which the function is one-to-one and non-decreasing, then find an inverse of the function on this domain. 1. f ( x) ( x 4). f ( x) ( x+ ). f ( x) 1 x 4. f ( x) 9 x 5. f ( x) x f ( x) 4 x Find the inverse of each function. 7. f ( x) 9+ 4x 4 8. f ( x) 6x f ( x) 9+ x 10. ( ) x f ( x) 1. f ( x) + x + x 7 1. f ( x) 14. f ( x) + x x 15. f ( x) 16. f ( x) f x x x 4 x x + 7 5x + 1 5x Police use the formula v 0L to estimate the speed of a car, v, in miles per hour, based on the length, L, in feet, of its skid marks when suddenly braking on a dry, asphalt road. 17. At the scene of an accident, a police officer measures a car's skid marks to be 15 feet long. Approximately how fast was the car traveling? 18. At the scene of an accident, a police officer measures a car's skid marks to be 15 feet long. Approximately how fast was the car traveling? The formula v.7r models the maximum safe speed, v, in miles per hour, at which a car can travel on a curved road with radius of curvature r, in feet. 19. A highway crew measures the radius of curvature at an exit ramp on a highway as 40 feet. What is the maximum safe speed? 0. A highway crew measures the radius of curvature at a tight corner on a highway as 900 feet. What is the maximum safe speed?

4 .5 Inverses and Radical Functions 1 1. A drainage canal has a crosssection in the shape of a parabola. Suppose that the canal is 10 feet deep and 0 feet wide at the top. If the water depth in the ditch is 5 feet, how wide is the surface of the water in the ditch? [UW]. Brooke is located 5 miles out from the nearest point A along a straight shoreline in her sea kayak. Hunger strikes and she wants to make it to Kono s for lunch; see picture. Brooke can paddle mph and walk 4 mph. [UW] a. If she paddles along a straight line course to the shore, find an expression that computes the total time to reach lunch in terms of the location where Brooke beaches her kayak. b. Determine the total time to reach Kono s if she paddles directly to the point A. c. Determine the total time to reach Kono s if she paddles directly to Kono s. d. Do you think your answer to b or c is the minimum time required for Brooke to reach lunch? e. Determine the total time to reach Kono s if she paddles directly to a point on the shore half way between point A and Kono s. How does this time compare to the times in parts b or c? Do you need to modify your answer to part d?. Clovis is standing at the edge of a dropoff, which slopes 4 feet downward from him for every 1 horizontal foot. He launches a small model rocket from where he is standing. With the origin of the coordinate system located where he is standing, and the x-axis extending horizontally, the path of the rocket is described by the formula y x + 10x. [UW] a. Give a function h f( x) relating the height h of the rocket above the sloping ground to its x-coordinate. b. Find the maximum height of the rocket above the sloping ground. What is its x-coordinate when it is at its maximum height? c. Clovis measures the height h of the rocket above the sloping ground while it is going up. Give a function x g( h) relating the x-coordinate of the rocket to h. d. Does the function from (c) still work when the rocket is going down? Explain.

5 x x y x+ 5 x 5. y 7. y 1( x ) ( x + 1) 7( x ) ( x+ )( x ) 6( x ) ( x + )( x ) ( x ) ( x+ 1)( x ) 4( x ) ( x+ )( x 4) 1( x+ )( x ) ( x ) ( x)( x ) ( x+ )( x 4) ( x 4)( x ) ( x 4)( x + 1) 7 x 4 x y x+ 4 x y 9. y 1. y. y 5. y 7. y 4 9. a. C( n) b. C ( 10) 1.% c. 80 ml d. as n, C n Section.5 1. Domain ( 4, ) Inverse ( ). Domain (,0) Inverse ( ) f x x Domain (, ) Inverse f ( x) f x 1 x x ( ) x 9 7. f ( x) f ( x) 1 x f 1 ( x) 8x 7x 1. f ( x) x x 5x f ( x) mph + 4x mph feet