Explanation Given: The functions f ( x ) = 1 x 2 + 1 and g ( x ) = x − 2 . Key concepts used: Composition of two functions is defined as: ⇒ f ∘ g ( x ) = f ( g ( x ) ) ⇒ g ∘ f ( x ) = g ( f ( x ) ) While finding domain of a composite function, restrictions on the composite function as well as restrictions on inner function are taken into account. Calculation: First of all, we find composite functions as shown below: ⇒ f ∘ g = f ( g ) ⇒ f ∘ g = f ( x − 2 ) ⇒ f ∘ g = 1 ( x − 2 ) 2 + 1 = 1 ( x − 2 ) 2 + 1 ⇒ f ∘ g = 1 x − 4 + 1 = 1 1 x 4 + 1 ⇒ f ∘ g = x 4 1 + x 4 This is a rational function without any restrictions as denominator of this function is never zero. So it doesn’t have any restrictions on the domain. The inner function is g ( x ) = x − 2 = 1 x 2

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 1.3, Problem 30E

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Composite functions f∘g and g∘f and their domains.

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Chapter 1.3, Problem 29E

Chapter 1.3, Problem 31E

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Composite functions f ∘ g and g ∘ f and their domains.

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Composite functions f∘g and g∘f and their domains.

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Chapter 1 Solutions