# calc 13 - montelongo(jcm3827 – Hw 13 – gualdani...

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Unformatted text preview: montelongo (jcm3827) – 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. graph of f passes horizontal line test; B. f is 1- 1; C. f ′ ( x ) ≥ 0 for all x . Which properties always ensure that f has an inverse f − 1 ? 1. B only 2. A only 3. A and C only 4. C only 5. B and C only 6. none of them 7. A and B only correct 8. all of them 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 ( f could be constant). 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 2. 4 8- 4- 8 4 8- 4- 8 montelongo (jcm3827) – Hw 13 – gualdani – (56455) 2 3. 4 8- 4- 8 4 8- 4- 8 correct 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 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 , ≤ x ≤ 6 correct 5. f − 1 ( x ) = 1 √ 36- x 2 , ≤ 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 montelongo (jcm3827) – Hw 13 – gualdani – (56455) 3 Find the inverse of f ( x ) = 1 + 4 x 8- 3 x . 1. f − 1 ( x ) = 8 x- 1 4 x + 3 2. f − 1 ( x ) = 8 x- 1 3 x + 4 correct 3. f − 1 ( x ) = 8 x + 1 3 x + 4 4. f − 1 ( x ) = 4 x- 1 3 x + 8 5. f − 1 ( x ) = 8 x + 1 4 x + 3 Explanation: To determine the inverse of f we first solve for x in y = 1 + 4 x 8- 3 x . In this case x = 8 y- 1 3 y + 4 . The inverse f − 1 is now obtained by inter- changing x, y . Thus y = f − 1 ( x ) = 8 x- 1 3 x + 4 . 005 10.0 points If a, b are the solutions of the exponential equation 2 x 2 = 4- 5 2 x +12 , calculate the value of ab ....
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calc 13 - montelongo(jcm3827 – Hw 13 – gualdani...

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