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hw 13 - silva(jrs4378 Hw 13 gualdani(56455 This print-out...

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silva (jrs4378) – Hw 13 – gualdani – (56455) 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 ) > 0 for all x ; B. f is 1 - 1; C. graph of f passes vertical line test. Which properties always ensure that f has an inverse f 1 ? 1. A only 2. none of them 3. A and C only 4. all of them 5. B and C only 6. A and B only correct 7. B only 8. C 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. True. C. False. 002 10.0 points If the graph of f is -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 4 8 - 4 - 8 4 8 - 4 - 8 which of the following is the graph of f 1 ( x )? 1. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 4 8 - 4 - 8 4 8 - 4 - 8 2. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 4 8 - 4 - 8 4 8 - 4 - 8
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silva (jrs4378) – Hw 13 – gualdani – (56455) 2 3. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 4 8 - 4 - 8 4 8 - 4 - 8 4. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 4 8 - 4 - 8 4 8 - 4 - 8 5. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 4 8 - 4 - 8 4 8 - 4 - 8 correct Explanation: The graph of f 1 ( x ) is obtained by reflect- ing the graph of f over the line y = x : -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 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 , 0 x 6 . 1. inverse doesn’t exist 2. f 1 ( x ) = radicalbig 36 - x 2 , - 6 x 6 3. f 1 ( x ) = radicalbig x 2 - 36 , - 6 x 6 4. f 1 ( x ) = radicalbig 36 - x 2 , 0 x 6 correct 5. f 1 ( x ) = 1 36 - x 2 , 0 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 , 0 x 6 . 004 10.0 points
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silva (jrs4378) – Hw 13 – gualdani – (56455) 3 Find the inverse of f ( x ) = 1 + 3 x 6 - 5 x . 1. f 1 ( x ) = 6 x - 1 3 x + 5 2. f 1 ( x ) = 6 x + 1 5 x + 3 3. f 1 ( x ) = 6 x + 1 3 x + 5 4. f 1 ( x ) = 6 x - 1 5 x + 3 correct 5. f 1 ( x ) = 3 x - 1 5 x + 6 Explanation: To determine the inverse of f we first solve for x in y = 1 + 3 x 6 - 5 x . In this case x = 6 y - 1 5 y + 3 . The inverse f 1 is now obtained by inter- changing x, y . Thus y = f 1 ( x ) = 6 x - 1 5 x + 3 . 005 10.0 points If a, b are the solutions of the exponential equation 5 x 2 = 25 5 x 12 , calculate the value of ab . 1. ab = 18 2. ab = - 24 3. ab = - 10 4. ab = 14 5. ab = 24 correct Explanation: By properties of exponents, 25 5 x 12 = 5 10 x 24 .
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