# HW 13 Calc - garcia (jjg2564) HW 13 gualdani (56410) 1 This...

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Unformatted text preview: garcia (jjg2564) HW 13 gualdani (56410) 1 This print-out should have 20 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. 001 10.0 points Consider the following properties that a func- tion f might have: A. f ( x ) &gt; 0 for all x ; B. graph of f passes vertical line test; C. f is 1- 1. Which properties always ensure that f has an inverse f 1 ? 1. none of them 2. A only 3. A and C only correct 4. C only 5. A and B only 6. B and C only 7. all of them 8. B only Explanation: A function f has an inverse, often denoted by f 1 , when there exists a function g so that g ( f ( x )) = x, f ( g ( x )) = x ; then f 1 = g . Such a function g exists if and only if (i) the graph of f passes the horizontal line test; (ii) the graph of f is strictly increasing or strictly decreasing; (iii) f is a 1-1 function. Consequently, A. True. B. False. C. True. 002 10.0 points If the graph of f is 4 8- 4- 8 4 8- 4- 8 which of the following is the graph of f 1 ( x )? 1. 4 8- 4- 8 4 8- 4- 8 correct 2. 4 8- 4- 8 4 8- 4- 8 garcia (jjg2564) HW 13 gualdani (56410) 2 3. 4 8- 4- 8 4 8- 4- 8 4. 4 8- 4- 8 4 8- 4- 8 5. 4 8- 4- 8 4 8- 4- 8 Explanation: The graph of f 1 ( x ) is obtained by reflect- ing the graph of f over the line y = x : 4 8- 4- 8 4 8- 4- 8 003 10.0 points Find the inverse function, f 1 , for f when f ( x ) = radicalbig 36- x 2 , x 6 . 1. inverse doesnt exist 2. f 1 ( x ) = 1 36- x 2 , x &lt; 6 3. f 1 ( x ) = radicalbig 36- x 2 , x 6 correct 4. f 1 ( x ) = radicalbig x 2- 36 ,- 6 x 6 5. f 1 ( x ) = radicalbig 36- x 2 ,- 6 x 6 Explanation: As the graph 6 6 f = f 1 of f shows, the given f is 1-1 and so has an inverse. In fact, the graph shows that f 1 = f , but to check this algebraically, we have to solve the equation x = radicalbig 36- y 2 for y . Squaring both sides and simplifying we obtain x 2 = 36- y 2 , i . e ., y 2 = 36- x 2 . Consequently, f 1 ( x ) = radicalbig 36- y 2 , x 6 . 004 10.0 points garcia (jjg2564) HW 13 gualdani (56410) 3 Find the inverse of f ( x ) = 1 + 2 x 8- 5 x . 1. f 1 ( x ) = 8 x + 1 5 x + 2 2. f 1 ( x ) = 8 x- 1 5 x + 2 correct 3. f 1 ( x ) = 2 x- 1 5 x + 8 4. f 1 ( x ) = 8 x + 1 2 x + 5 5. f 1 ( x ) = 8 x- 1 2 x + 5 Explanation: To determine the inverse of f we first solve for x in y = 1 + 2 x 8- 5 x . In this case x = 8 y- 1 5 y + 2 . The inverse f 1 is now obtained by inter- changing x, y . Thus y = f 1 ( x ) = 8 x- 1 5 x + 2 . 005 10.0 points If a, b are the solutions of the exponential equation 5 x 2 = 25 x +12 , calculate the value of ab ....
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## This note was uploaded on 10/08/2010 for the course MATH 408K taught by Professor Gualdani during the Fall '09 term at University of Texas at Austin.

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HW 13 Calc - garcia (jjg2564) HW 13 gualdani (56410) 1 This...

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