EXAM1.pdf - Version 003 EXAM1 villafuerte altu(53780 This...

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Version 003 – EXAM1 – villafuerte altu – (53780) 1 This print-out should have 19 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0points Find the inverse function, f 1 , of f ( x ) = x 2 5 ( x 0) . 1. f 1 ( x ) = radicalbigg x 5 , ( x 0) 2. f 1 ( x ) = 5 x, ( x 0) 3. f 1 ( x ) = 5 x, ( x 0) 4. f 1 ( x ) = radicalbigg x 5 , ( x 0) 5. f 1 ( x ) = radicalbigg x 5 , ( x 0) 6. f 1 ( x ) = 5 x, ( x 0) correct Explanation: The graph of f ( x ) = 1 5 x 2 y x as a function on ( −∞ , ) fails the horizontal line test, and f will not have an inverse. But if we restrict the domain of f ( x ) to [0 , ), then the graph will pass the horizontal line test. In this case f 1 ( x ) will exist. Also, because then domain f = ( −∞ , 0] , range f = [0 , ) , we see that domain f 1 = [0 , ) , range f 1 = [0 , ) . To determine f 1 we interchange x and y in y = f ( x ) = 5 x 2 , and solve for y : y 2 = 5 x, i.e., y = ± 5 x. But which square root do we take? Well, the range of f 1 is [0 , ), so this means we must take the positive square root. Consequently, f 1 ( x ) = 5 x, ( x 0) . 002 10.0points Which of the following is the graph of f ( x ) = 2 2 x 1 ? 1. 2 4 2 4 2 4 2 4 2. 2 4 2 4 2 4 2 4
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Version 003 – EXAM1 – villafuerte altu – (53780) 2 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 correct 6. 2 4 2 4 2 4 2 4 Explanation: Since lim x → ∞ 2 x = 0 , we see that lim x → ∞ f ( x ) = 2 , in particular, f has a horizontal asymptote y = 2. This eliminates all but two of the graphs. On the other hand, f (0) = 3 2 , so the y -intercept of the given graph must occur at y = 3 2 . Consequently, the graph is of f is 2 4 2 4 2 4 2 4 003 10.0points Below is the graph of a function f . 2 4 6 2 4 6 2 4 6 8 2 4 Use the graph to determine lim x → − 4 f ( x ). 1. lim x → − 4 f ( x ) = 6 2. lim x → − 4 f ( x ) does not exist 3. lim x → − 4 f ( x ) = 4 4. lim x → − 4 f ( x ) = 8
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Version 003 – EXAM1 – villafuerte altu – (53780) 3 5. lim x → − 4 f ( x ) = 2 correct Explanation: From the graph it is clear the f has both a left hand limit and a right hand limit at x = 4; in addition, these limits coincide. Thus lim x → − 4 f ( x ) = 2 . 004 10.0points Consider the function f ( x ) = 1 x, x< 1 x, 1 x< 2 ( x 2) 2 , x 2 . Find all the values of a for which the limit lim x a f ( x ) exists, expressing your answer in interval no- tation. 1. ( −∞ , 1) ( 1 , ) 2. ( −∞ , 1) ( 1 , 2) (2 , ) correct 3. ( −∞ , ) 4. ( −∞ , 1] [2 , ) 5. ( −∞ , 2) (2 , ) Explanation: The graph of f is a straight line on ( −∞ , 1), so lim x a f ( x ) exists (and = f ( a )) for all a in ( −∞ , 1).
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