Q7SOL - MAP 103 Prociency Algebra Quiz#7 Fall 2009...

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MAP 103: Proficiency Algebra, Quiz #7, Fall 2009 SOLUTIONS 1. Let f ( x ) = 2 + 3 x x + 1 . Find a formula for the inverse f - 1 ( x ) and solve the equation f - 1 ( x ) = 2 . Sol. We first find the inverse of f ( x ) . A computation follows. f ( x ) = 2 + 3 x x + 1 y = 2 + 3 x x + 1 y ( x + 1) = 2 + 3 x yx + y = 2 + 3 x yx - 3 x = 2 - y x ( y - 3) = 2 - y x = 2 - y y - 3 f - 1 ( x ) = 2 - x x - 3 We next solve the equation f - 1 ( x ) = 2 . A computation follows. f - 1 ( x ) = 2 2 - x x - 3 = 2 2 - x = 2( x - 3) 2 - x = 2 x - 6 6 + 2 = 2 x + x 8 = 3 x x = 8 3 2. Let g ( x ) = 1 2 x - 1 2 . Find a formula for the inverse g - 1 ( x ) and show that g - 1 g ( x ) = x. What is the domain for g - 1 ( x )? Sol. We first find the inverse. A computation follows. g ( x ) = 1 2 x - 1 2 y = 1 2 x - 1 2 x = 1 2 y - 1 2 2 x = y - 1 y = 2 x + 1 g - 1 ( x ) = 2 x + 1
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We next verify that g - 1 g ( x ) = x. A computation follows. g - 1 g ( x ) = x (2 x + 1) (1 / 2 x - 1 / 2) = x 2 1 2 x - 1 2 + 1 = x x - 1 + 1 = x x = x It follows the domain of the inverse is D ( g - 1 ( x )) = ( -∞ , ) . 3. Let h ( x ) = e x + 2 . By reflecting the graph of h ( x ) over the line f ( x ) = x, sketch a graph for the inverse h - 1 ( x ) of h ( x ) . Sol. 4.
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