quiz4aANS - that f a n = b for any a ∈ R As an example if...

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Math 240, Quiz 4 Name: Circle One: T 12:05 T 2:25 R 12:05 R 2:25 Instructions: Answer all questions fully, showing work where necessary. 1)Determine whether the following function from R to R are one-to-one and/or onto. f ( x ) = - 3 x 2 + 7 To see if it is one-to-one, we need to see that if f ( a ) = f ( b ), then a = b . Thus we suppose f ( a ) = f ( b ), or - 3 a 2 +7 = - 3 b 2 +7. Subtracting seven, we see that - 3 a 2 = - 3 b 2 . Dividing by - 3, we get a 2 = b 2 . But this does not imply that a = b . For example, since 1 2 = ( - 1) 2 , but 1 n = - 1 (in fact, f (1) = 4 = f ( - 1). Thus, this function is not one-to-one. To see if it is onto, we let b be a real number, and see if we can ±nd a such that f ( a ) = b . In other words, we want - 3 a 2 + 7 = b , or - 3 a 2 = b - 7, or a 2 = b - 7 - 3 . But if b - 7 > 0, then the right side is negative. But a 2 0. So if b - 7 > 0, or b > 7, we have
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Unformatted text preview: that f ( a ) n = b for any a ∈ R . As an example, if b = 8, then we would need that a 2 =-1 3 . 2) Find the least integer n such that f ( x ) is O ( x n ) for each of these func-tions. a) f ( x ) = 2 x 2 + x 3 log x 2 x 2 is O ( x 2 ), while x 3 log x is O ( x 3 log x ). But log x < x , so x 3 log x is O ( x 4 ). Thus, when we add these two functions, we get that f ( x ) is O (max( x 2 ,x 4 )) = 0( x 4 ). Thus, the answer is n = 4. b) f ( x ) = ( x 3 + 5 log x ) / ( x 4 + 1) We can say the top is O ( x 3 ), while the bottom is O ( 1 x 4 ). So when you multiply the two functions together, you get O ( 1 x ). 1...
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This note was uploaded on 08/11/2008 for the course MATH 240 taught by Professor Miller during the Fall '08 term at University of Wisconsin.

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