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# 73 - Dept of Mathematics The City College of New York...

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Dept of Mathematics The City College of New York Mathematics 19500 Final Examination Spring 2005 Instructions: Solve any 10 complete problems. Omit two problems. Show all work. Calculators are not permitted Leave your answers in terms of π , , e , ln , and fractions. 1) Let 1 ( ) 2 3 x x + = f x and 2 1 ) ( x x g = a) Find and simplify . )) 3 ( ( x g g b) Find the inverse function . 1 ( ) f x 2) Write each answer in interval form: a) Solve the inequality . 2 3 11 18 x x x + b) Find the domain of the function x x x f 4 9 ) ( = . 3) a) Sketch the graph of . Find all intercepts and the equation of the asymptote. 4 2 + = x e y b) Find an equation of the straight line through the point (5 that is perpendicular to the line with equation 3 . , 4) 2 8 x y + = 4) Find all intercepts and asymptotes, and sketch the graph of the function 2 2 6 1 x x y x + = . 5) Prove the identity 1 1 csc csc tan 2 2 2 = x x x . 6) For the function , ) 3 sin( 2 π + = x y a) Find the amplitude, period, and phase shift.

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