HW 13 Gagan Solutions - Gauss, Karl Friedric Homework 13 R...

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Gauss, Karl Friedric – Homework 13 – R Gompf – 57965 – Nov 10, 2004 1 This print-out should have 22 questions. Multiple-choice questions may continue on the next column or page – ±nd all choices before answering. The due time is Central time. 001 (part 1 of 1) 10 points Five functions are de±ned by the table x f ( x ) g ( x ) h ( x ) F ( x ) G ( x ) 123456 382687 464791 513467 435491 245787 Which of these functions is one-to-one? 1. G ( x ) 2. g ( x ) 3. h ( x ) correct 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) . 002 (part 1 of 1) 10 points Which of the following functions will fail to have an inverse? 1. 24 2 4 2 4 2 4 correct 2. 2 4 2 4 2 4 3. 2 4 2 4 2 4 4. 2 4 2 4 2 4 5. 2 4 2 4 2 4 Explanation:
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Gauss, Karl Friedric – Homework 13 – R Gompf – 57965 – Nov 10, 2004 2 If a function f has an inverse, then it must be a one-to-one function; equivalently, the graph of f must pass the horizontal line test. The only graph failing to pass this test is 24 2 4 2 4 2 4 003 (part 1 of 1) 10 points Find a formula for the inverse of the function f ( x )=3 x +7 . 1. f 1 ( x )= x +3 7 2. f 1 ( x x 3 7 3. f 1 ( x x 4. f 1 ( x x 7 3 correct 5. f 1 ( x x 3 Explanation: y = f ( x x 3 x =7 y x = 7 y 3 . Interchange x and y: y = 7 x 3 . So f 1 ( x x 7 3 004 (part 1 of 1) 10 points Find the inverse function, f 1 ,of f ( x 3 x 5 x 2 +9 . 1. f 1 ( x 3 x 9+5 x 2 2. f 1 ( x 3 x 9 5 x 2 correct 3. f 1 ( x 9 9 x 2 5 4. f 1 ( x x 2 3 x 5. f 1 ( x 9 5 x 2 5 x 6. f 1 ( x 5 9 9 x 2 Explanation: The graph of f ( x 3 x 5 x 2 , is shown as a continuous line in f f 1 Since this graph passes the horizontal line test, f will have an inverse, f 1 , whose graph is shown as a dotted line. Algebraically, this inverse function is de±ned by y = f 1 ( x ) where x = 3 y p 5 y 2 . Solving for y we see that 9 y 2 = x 2 ( 5 y 2 ) =5 x 2 y 2 x 2 ,
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Gauss, Karl Friedric – Homework 13 – R Gompf – 57965 – Nov 10, 2004 3 i.e. , y 2 ( 9 5 x 2 ) =9 x 2 . Thus y = 3 x 9 5 x 2 , 3 5 <x< 3 5 , and so f 1 ( x )= 3 x 9 5 x 2 , 3 5 3 5 . 005 (part 1 of 1) 10 points When g is the inverse of the function f ( x x 3 +5 x 4 , ±nd the value of g 0 (14). ( Hint : f (2) = 14.) 1. g 0 (14) = 4 17 2. g 0 (14) = 17 3. g 0 (14) = 2 17 4. g 0 (14) = 17 2 5. g 0 (14) = 17 4 6. g 0 (14) = 1 17 correct Explanation: By de±nition, g ( f ( x )) = x, so by the Chain Rule, g 0 ( f ( x )) f 0 ( x )=1 . Thus g 0 ( f ( x )) = 1 f 0 ( x ) . To determine the value of g 0 ( a ), therefore, we need to know b so that f ( b a ,forthen g 0 ( a 1 f 0 ( b ) .
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HW 13 Gagan Solutions - Gauss, Karl Friedric Homework 13 R...

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