Unformatted text preview: g ( f ) is the same as the domain of f , the set of all real numbers x such that x 2 = 3. C01S04.014: If f ( x ) = x 2 + 1 and g ( x ) = 1 x 2 + 1 , then f ( g ( x )) = ( g ( x )) 2 + 1 = 1 ( x 2 + 1) 2 + 1 = x 4 + 2 x 2 + 2 x 4 + 2 x 2 + 1 and g ( f ( x )) = 1 ( f ( x )) 2 + 1 = 1 ( x 2 + 1) 2 + 1 = 1 x 4 + 2 x 2 + 2 . C01S04.015: If f ( x ) = x 3 − 4 and g ( x ) = ( x + 4) 1 / 3 , then f ( g ( x )) = ( g ( x )) 3 − 4 = ³ ( x + 4) 1 / 3 ´ 3 − 4 = x + 4 − 4 = x and g ( f ( x )) = ( f ( x ) + 4) 1 / 3 = ( x 3 − 4 + 4 ) 1 / 3 = ( x 3 ) 1 / 3 = x. The domain of both f ( g ) and g ( f ) is the set R of all real numbers, so here is an example of the highly unusual case in which f ( g ) and g ( f ) are the same function. C01S04.016: If f ( x ) = √ x and g ( x ) = cos x , then 2...
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 Spring '11
 CHENG
 Calculus, Real Numbers

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