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assignment1 - ma2176 a1 1 Functions Limits and Continuity...

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1 ma2176 a1 Functions, Limits and Continuity 1. Find the largest possible domain of each of the following functions: (a) 5 4 2 ) ( 2 = = x x x x f y , (b) 2 25 ) ( x x f y = = , (c) 1 1 ) ( 2 = x x x φ 2. Let 3 2 ( ) 2, ( ) 1 f x x g x x = + = . Find formulas for (a) ( ) ( ) f g x + , (b) ( ) g x f , (c) ( ) ( ) g f x D , (d) ( ) ( ) f g x D and state their largest possible domains. 3. State whether each of the following is an odd function, an even function or neither. Prove your statements or give counterexamples. (a) The sum of two even functions. (b) The sum of two odd functions. (c) The product of two even functions. (d) The product of two odd functions. (e) The product of an even function and an odd function. 4. Let F be any function whose domain contains x whenever it contains x . Prove each of the following: (a) ) ( ) ( x F x F determines an odd function. (b) ) ( ) ( x F x F + determines an even function. (c) F can always be expressed as the sum of an odd function and an even function.
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