Homework 1

# Homework 1 - b (0) = − 2 , b (1) = 0 , b (2) = 4 , b (3)...

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INSTRUCTOR: JOS ´ E MANUEL G ´ OMEZ Homework 1. Due date Wednesday September 7, 2011 Please show all your work. (1) (5 Points) The functions f and g are deFned below f ( x ) = x 2 1 , g ( x ) = 1 x . ±ind explicit descriptions and the domains of the following functions. (a) f g , (b) g f , (c) f f , (d) g g . (2) (5 Points) Consider the functions p ( x ) = x 3 x x + 1 , q ( x ) = x 2 x. (a) ±ind the domains of p ( x ) and q ( x ). (b) Show that p ( x ) = q ( x ) for all x n = 1. (c) Are p ( x ) and q ( x ) the same functions? Please explain your answer. (d) Provide a sketch for the graph of p ( x ). (3) (5 Points) Let h be function deFned by h ( x ) = x 1 x + 1 . (a) Show that h is a one to one function. (b) ±ind the inverse function of h . (4) (5 Points) Solve the following equations (a) ln( x 2 1) ln( x + 1) = 1 (b) e ln( x +1) - ln( x - 1) = x (5) (5 Points) Let a be the function deFned by a ( x ) = cos( πx 4 ). Assume also that b is a one to one function such that

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Unformatted text preview: b (0) = − 2 , b (1) = 0 , b (2) = 4 , b (3) = 5 and let c be the function whose graph is given below 1 2 INSTRUCTOR: JOS ´ E MANUEL G ´ OMEZ-3-2-1 1 2 3-2-1 1 2 3 ◦ y = c ( x ) x y Find the following quantities (a) a ◦ b-1 (0) . (b) b ◦ c (2). (c) c ◦ a (4). (6) (5 Points) Radioactive isotopes are used in real life to determine the absolute age of fossils. The amount of a given radioactive isotope W ( t ) present in a fossil after t years, is given by the equation W ( t ) = W e-λt . In this case λ = 0 . 0023104 and W is the initial quantity of the radioactive isotope. Find the half life of this isotope. Recall that the half life is de±ned to be the time that it takes for half of the radioactive material to decay....
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## This note was uploaded on 01/20/2012 for the course CALC AS.110.106 taught by Professor Josegomez during the Fall '11 term at Johns Hopkins.

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Homework 1 - b (0) = − 2 , b (1) = 0 , b (2) = 4 , b (3)...

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