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Solutions Homework 1

# Solutions Homework 1 - MATH 106 CALCULUS I FOR BIO SOC SCI...

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MATH 106 CALCULUS I FOR BIO. & SOC. SCI. FALL 2011 INSTRUCTOR: JOS ´ E MANUEL G ´ OMEZ Solutions Homework 1. Please show all your work. (1) (5 Points) The functions f and g are defined below f ( x ) = x 2 1 , g ( x ) = 1 x . Find explicit descriptions and the domains of the following functions. (a) f g , (b) g f , (c) f f , (d) g g . Solution: Let’s start by finding the domains of f and g . Clearly the domain of g are all numbers x negationslash = 0. On the other hand, the domain of f is the set of numbers x such that x 2 1 0; that is x 2 1. This shows that | x | = x 2 1. The solution of this inequality is the set of elements x 1 and x ≤ − 1. (a) f g ( x ) = f ( g ( x )) = f parenleftbigg 1 x parenrightbigg = radicalbigg 1 x 2 1 . The domain of f g is the set of elements x for which g ( x ) is defined and g ( x ) is in the domain of f . This shows that x negationslash = 0 and that | g ( x ) | = 1 | x | 1. This shows that | x | ≤ 1. It follows that the domain of f g is [ 1 , 0) (0 , 1]. (b) g f ( x ) = g ( f ( x )) = g parenleftBig x 2 1 parenrightBig = 1 x 2 1 . The domain of g f is the set of elements for which f ( x ) = x 2 1 makes sense and f ( x ) is in the domain of g ; that is, f ( x ) negationslash = 0. So we need that x 1 or x ≤ − 1 and we also need x 2 1 negationslash = 0. Note that x 2 1 = 0 precisely when x 2 = 1; that is, x = ± 1. We conclude that the domain of g f is ( −∞ , 1) (1 , ).

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Solutions Homework 1 - MATH 106 CALCULUS I FOR BIO SOC SCI...

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