HW12-solutions

HW12-solutions - husain(aih243 – HW12 – Gilbert...

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Unformatted text preview: husain (aih243) – HW12 – Gilbert – (56215) 1 This print-out should have 26 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Five functions are defined by the table x f ( x ) g ( x ) h ( x ) F ( x ) G ( x ) 1 2 3 4 5 6 2 3 5 9 3 1 7 1 7 4 8 2 5 8 9 6 3 2 9 3 7 9 8 2 4 3 6 9 1 9 Which of these functions is one-to-one? 1. h ( x ) correct 2. g ( x ) 3. G ( x ) 4. f ( x ) 5. F ( x ) Explanation: The function y = h ( x ) is the only one-to- one function since f (2) = f (5) , g (1) = g (3) , F (1) = F (4) , G (4) = G (6) . keywords: one-to-one, function value, 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 3. 4 8- 4- 8 4 8- 4- 8 husain (aih243) – HW12 – Gilbert – (56215) 2 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 Which of the following functions will fail to have an inverse? 1. 2 4- 2- 4 2 4- 2- 4 correct 2. 2 4- 2- 4 2 4- 2- 4 3. 2 4- 2- 4 2 4- 2- 4 4. 2 4- 2- 4 2 4- 2- 4 5. 2 4- 2- 4 2 4- 2- 4 Explanation: If a function f has an inverse, then it must be a one-to-one function; equivalently, the husain (aih243) – HW12 – Gilbert – (56215) 3 graph of f must pass the horizontal line test. The only graph failing to pass this test is 2 4- 2- 4 2 4- 2- 4 004 10.0 points If f is a one-to-one function such that f (3) = 1, what is the value of f − 1 (1)? 1. f − 1 (1) = 3 correct 2. not enough information given 3. f − 1 (1) = 9 4. f − 1 (1) = 2 5. f − 1 (1) = 5 6. f − 1 (1) = 7 Explanation: Since f (3) = 1 and f is one-to-one, the inverse, f − 1 , of f exists and f − 1 ( f ( x )) = f ( f − 1 ( x )) = x . Consequently, f − 1 (1) = f − 1 ( f (3)) = 3 . keywords: inverse, one-to-one, 005 10.0 points Determine if f ( x ) = 8- 6 x has an inverse, and find this inverse if it does. 1. f − 1 ( x ) does not exist 2. f − 1 ( x ) = 1 6 x- 4 3 3. f − 1 ( x ) = 4 3- 1 6 x correct 4. f − 1 ( x ) = 1 8- 6 x 5. f − 1 ( x ) = 4 3 + 1 6 x Explanation: A linear function with non-zero slope is either an increasing function or a decreasing function, depending on the sign of the slope. In either case, however, it is a 1-1 function, and so has an inverse. To determine this inverse when f ( x ) = 8- 6 x we first solve for x in y = 8- 6 x : x = 4 3- 1 6 y , and then interchange x, y . Thus f − 1 ( x ) = 4 3- 1 6 x . 006 10.0 points Find the inverse function, f − 1 , for f when f ( x ) = radicalbig 25- x 2 , ≤ x ≤ 5 ....
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This note was uploaded on 11/30/2010 for the course M 408c taught by Professor Mcadam during the Spring '06 term at University of Texas.

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HW12-solutions - husain(aih243 – HW12 – Gilbert...

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