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assignment1bfsol

# assignment1bfsol - ma2176 a1 Brief solution 1 Functions...

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1 ma2176 a1 Brief solution 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 Solution : (a) ( )( ) 5 or 1 0 1 5 5 4 2 = = = + = x x x x x x . The largest domain of ) ( x f { } \ 1,5 R = (b) ( )( ) 5 5 0 5 5 0 25 2 + x x x x . The largest domain of ) ( x f ] 5 , 5 [ = (c) 1 0 1 = = x x . The largest domain of ) ( x φ { } \ 1 R = 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. Solution : (a) ( ) ( ) 1 2 2 ) )( ( 3 + + = + = + x x x g x f x g f Since the largest domain of ) ( x f R = and the largest domain of ) ( x g { } \ 1 R = , the largest domain of ) )( ( x g f + is { } { } \ 1 \ 1 R R R = . (b) ( ) ( ) ( ) ) 2 ( 1 2 2 1 2 ) ( 3 3 + = + = = x x x x x f x g x f g Since the largest domain of ) ( x f R = , ( ) { } { } ( ) { } 3 / 1 3 2 0 2 : 0 : = = + = = x x x f x and the largest domain of ) ( x g { } \ 1 R = , the largest domain of ) ( x f g is ( ) { } ( ) { } ( ) ( ) { } 1/3 1/3 \ 2 \ 1 \ 1, 2 R R R = . (c) ( ) ( ) ( ) 1 2 1 2 2 2 ) )( ( 3 3 3 + = + = + = = x x x g x f g x f g D The largest domain of ( ) ( ) 3 2 ( )( ) 1 g f x g f x x = = + D is \{ 1} R . (d) ( ) ( ) 2 1 2 1 2 ) )( ( 3 + = = = x x f x g f x g f D The largest domain of 2 1 2 ) )( ( 3 + = x x g f D is \{1} R .

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2 3. State whether each of the following is an odd function, an even function or neither. Prove your statements or give counterexamples.
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assignment1bfsol - ma2176 a1 Brief solution 1 Functions...

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