Lect18 - Composition: Consider the function f ( x )= x + 1...

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92.131 Lecture 18 1 of 12 Ronald Brent © 2010 All rights reserved. Composition: Consider the function 1 ) ( + = x x f . Then 1 ) 0 ( = f , 2 ) 1 ( = f , 1 ) ( 4 4 + = x x f , 1 sin ) sin ( + = x x f , 1 ) ( ) ) ( ( + = x g x g f , a n d 1 whatever ) whatever ( + = f . In diagram, x ) ) ( ( x g f ) ( x g ) ( x f
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92.131 Lecture 18 2 of 12 Ronald Brent © 2010 All rights reserved. Example: Let x x x f 2 ) ( 3 + = and 4 ) ( x x g = . Then what is ( ) ) ( x g f ? () ( ) ) ( 2 ) ( ) ( 3 x g x g x g f + = 4 12 4 3 4 2 2 ) ( x x x x + = + = Example: If x x f sin ) ( = and 1 ) ( + = x x g , then what is ( ) ) ( x g f and ( ) ) ( x f g ? a ) () ( ) 1 sin ) ( sin ) ( + = = x x g x g f . b ) () 1 sin 1 ) ( ) ( + = + = x x f x f g . Notice that ( ) ) ( x g f ( ) ) ( x f g . Definition: () ) ( x g f is called the composition of f with g . Its symbol is g f o , and its evaluation at a point x is denoted ) ( ) ( x g f o . Example: Let 2 1 ) ( x x f = , and x x g sec ) ( = . Find ) ( ) ( x g f o . Solution: () () x x x g x g f x g f 2 2 2 cos sec 1 ) ( 1 ) ( ) ( ) ( = = = =
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92.131 Lecture 18 3 of 12 Ronald Brent © 2010 All rights reserved. Decomposition In the composition, ( ) ) ( x g f , f is called the outer function, and g is called the inner function. Taking derivatives of functions that are composites requires using the chain rule. It is important to let f be the outermost function (because there may be several ways of decomposing). How can you find the outermost function? Answer: if you were to evaluate the function at some point, the last operation you would do corresponds to the outermost function. Example:
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This note was uploaded on 02/13/2012 for the course MATH 92.131 taught by Professor Staff during the Fall '09 term at UMass Lowell.

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Lect18 - Composition: Consider the function f ( x )= x + 1...

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