# Exam3 5 - But if f x = 3 x 3-1 x we know that f x = 9 x 2 1...

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Betancourt, Daniel – Exam 3 – Due: May 1 2007, 11:00 pm – Inst: Diane Radin 5 The function f ( x ) = 3 x is a one-to-one function for x 0; hence the inverse f - 1 ( x ) exists. Now the solution to the equation 3 x = y , y > 0 is x = 1 3 y 2 . But the original function f ( x ) can take only non-negative values, i.e. , the range of f is [0 , ); this means that the domain of f - 1 is [0 , ). Consequently, f - 1 ( x ) = 1 3 x 2 , ( x 0) . keywords: inverse function, algebraic 006 (part 1 of 1) 10 points Find the value of g 0 (2) when g is the inverse of the function f de±ned by f ( x ) = 3 x 3 - 1 x . ( Hint : f (1) = 2.) 1. g 0 (2) = 1 10 correct 2. g 0 (2) = - 10 3. g 0 (2) = - 3 4. g 0 (2) = 3 5. g 0 (2) = - 1 10 6. g 0 (2) = 10 Explanation: When g is the inverse of f then f ( g ( x )) = x, so f 0 ( g ( x )) g 0 ( x ) = 1 , i.e. , g 0 ( x ) = 1 f 0 ( g ( x )) .
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Unformatted text preview: But, if f ( x ) = 3 x 3-1 x , we know that f ( x ) = 9 x 2 + 1 x 2 , f (1) = 2 ; in particular, g (2) = 1 and f (1) = 10 . Consequently, g (2) = 1 f (1) = 1 10 . keywords: 007 (part 1 of 1) 10 points Find the value of lim x →-∞ µ 5 e x + e-x 2 e x-5 e-x ¶ . 1. limit = 1 5 2. limit = 5 3. limit =-4 7 4. limit =-1 5 correct 5. limit =-5 6. limit = 4 7 Explanation: Multiplying top and bottom by e x we see that 5 e x + e-x 2 e x-5 e-x = 5 e 2 x + 1 2 e 2 x-5 . On the other hand, lim x →-∞ e ax = 0...
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## This note was uploaded on 02/14/2012 for the course MATH 408 K taught by Professor Clark,c.w./hoy,r.r during the Spring '08 term at University of Texas.

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