30_Cal_Solution of Calculus_6e

30_Cal_Solution of Calculus_6e - g ( f ) is the same as the...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
C01S04.010: The graph of g ( x ) = sin10 x resembles that of the sine function, but with much more “activity” because of the factor 10. Multiply by the rapidly decreasing positive numbers 2 x and you will see the sine oscillations decreasing from the range [ 1 , 1] when x is near zero to very small oscillations—near zero—as x increases. So the graph of f is the one shown in Fig. 1.4.30. C01S04.011: Given f ( x )=1 x 2 and g ( x )=2 x + 3, f ( g ( x )) = 1 ( g ( x )) 2 =1 (2 x + 3) 2 = 4 x 2 12 x 8 and g ( f ( x )) = 2 f ( x ) + 3 = 2(1 x 2 )+3= 2 x 2 +5 . C01S04.012: Given f ( x )= 17 and g ( x )= | x | , f ( g ( x )) = 17 and g ( f ( x )) = | f ( x ) | = |− 17 | =17 . The ±rst result is a little puzzling until one realizes that to obtain f ( g ( x )), one substitutes g ( x ) for x for every occurrence of x in the formula for f .No x there means there’s no place to put g ( x ). Indeed, f ( h ( x )) = 17 no matter what the formula of h . C01S04.013: If f ( x )= x 2 3 and g ( x )= x 2 + 3, then f ( g ( x )) = p ( g ( x )) 2 3= p ( x 2 + 3) 2 3= p x 4 +6 x 2 + 6 and g ( f ( x )) = ( f ( x )) 2 +3= ³ p x 2 3 ´ 2 +3= x 2 3+3= x 2 . The domain of f ( g ) is the set R of all real numbers, but the domain of
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online