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

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Unformatted text preview: MAP 103: Proficiency Algebra, Quiz #7, Fall 2009 SOLUTIONS 1. Let f (x) = 2 + 3x . Find a formula for the inverse f −1 (x) and solve the equation f −1 (x) = 2. x+1 Sol. We first find the inverse of f (x). A computation follows. 2 + 3x x+1 2 + 3x y= x+1 y (x + 1) = 2 + 3x yx + y = 2 + 3x yx − 3x = 2 − y x(y − 3) = 2 − y 2−y x= y−3 2−x f −1 (x) = x−3 f (x) = We next solve the equation f −1 (x) = 2. A computation follows. f −1 (x) = 2 2−x =2 x−3 2 − x = 2(x − 3) 2 − x = 2x − 6 6 + 2 = 2x + x 8 = 3x 8 x= 3 2. Let g (x) = 1 x − 1 . Find a formula for the inverse g −1 (x) and show that g −1 ◦ g (x) = x. What 2 2 is the domain for g −1 (x)? Sol. We first find the inverse. A computation follows. 1 1 g (x) = x − 2 2 1 1 y = x− 2 2 1 1 x= y− 2 2 2x = y − 1 y = 2x + 1 g − 1 ( x) = 2 x + 1 We next verify that g −1 ◦ g (x) = x. A computation follows. g − 1 ◦ g ( x) = x (2x + 1) ◦ (1/2x − 1/2) = x 1 1 2 x− +1=x 2 2 x−1+1=x x=x It follows the domain of the inverse is D(g −1 (x)) = (−∞, ∞). 3. Let h(x) = ex + 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. Let f (x) = x2 + 1. Graph the function g (x) = −2f (x + 1) + 2. Sol. √ 5. Let f (x) = x + 1 and g (x) = 3x + 2. Find the composition g ◦ f and find the value of f ◦ g (2). Sol. We first find the composition g ◦ f. A computation follows. √ = (3x + 2) ◦ ( x + 1) √ = 3( x + 1) + 2 We next find f ◦ g (2). To do this we must note that we need to find g (2) first.√If x = 2, then √ g (2) = 3(2) + 2 = 6 + 2 = 8. Finally, we find f (g (2)) = f (8) = 8 + 1 = 9 = 3. Thus, f ◦ g (2) = 3. =g◦f Bonus (+10 Points): Solve for x : 27x Sol. 2 +1 = 243. 2 +1 27x 3 = 243 = 35 3(x2 +1) 3(x2 + 1) = 5 3x2 + 3 = 5 3x2 = 2 2 x2 = 3√ x=± 6 3 ...
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This note was uploaded on 04/18/2010 for the course MAP 103 taught by Professor Staff during the Fall '08 term at SUNY Stony Brook.

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

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