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Fall 2006 - Exam 1 (Version 1)

Fall 2006 - Exam 1 (Version 1) - Math 20A Midterm Exam...

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Math 20A Midterm Exam 1 (version 1) Solution 1. (6 points) Let f ( x ) = e 2 x and g ( x ) = 2 ln( x ). (a) Find the domain and range of f ( x ). The domain of f is R , the set of all real numbers and the range of f is (0 , ), the set of all positive real numbers. (b) Does f ( x ) have an inverse function? Justify your answer. Yes, f ( x ) has an inverse function because it is an increasing function, hence one-to-one. (c) Are the functions f ( x ) and g ( x ) inverses of each other? Explain how you determined your answer. No: g ( x ) is not the inverse of f ( x ) since f ( g ( x )) = e 2 g ( x ) = e 2 · 2ln( x ) = x 4 negationslash = x . (Note: it would be just as good to observe that g ( f ( x )) negationslash = x .) 2. (6 points) For each of the following limits, either evaluate it or explain why it does not exist. (a) lim x 0 braceleftbigg 1 2 x - 1 x ( x + 2) bracerightbigg lim x 0 1 2 x 1 x ( x + 2) ff = lim x 0 ( x + 2) 2 2 x ( x + 2) = lim x 0 x 2 x ( x + 2) = lim x 0 1 2( x + 2) = 1 4 (b) lim x 7 2 x 2 x - 7 lim x 7 2 x 2 x 7 does not exist because lim x 7 + 2 x 2 x 7 = . (Note that lim x
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