composition

composition - Composition of functions Given two functions...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Composition of functions Given two functions f and g, the composite function h = f o g is the function that consists of the action of g followed by the action of f and it is defined by = ( ) h x ( ) f ( ) g x , that is, ( f o g ) = ( ) x ( ) f ( ) g x . The domain of f o g is the subset of the domain of g consisting of those numbers x such that ( ) g x is in the domain of f, that is, D( f o g ) = { x | x D( g ) and ( ) g x D( f ) }. We need to restrict the domain of g to ensure that all output numbers from g are in the domain of f, so that f can then be applied to them. The following diagram illustrates the situation schematically. The blue shaded region represents R( g ) D( f ). Example 1 : Let the function f be given by = ( ) f x + x 3 and g be given by = ( ) g x x 2 . Find (a) (f o g) ( ) 5 (b) (f o g) ( ) u (c) (f o g) ( ) x (d) (g o f) ( ) x . Solution : (a) (f o g) = ( ) 5 ( ) f ( ) g 5 = ( ) f 25 = 28 (b) (f o g) = ( ) u ( ) f ( ) g u = ( ) f u 2 = + u 2 3 (c) (f o g) = ( ) x ( ) f ( ) g x = ( ) f x 2 = + x 2 3 , that is, (f o g) = ( ) x + x 2 3. Note : The action of the function f is to add 3 to any input number, while the action of g is to square any input number. The action of f o g is to perform both operations in the requisite order, namely to square any input number and then add 3 to the result. (d) (g o f) = ( ) x ( ) g ( ) f x = ( ) g + x 3 = ( ) + x 3 2 , that is, (g o f) = ( ) x ( ) + x 3 2 . Example 2 : Let the function f be given by = ( ) f x x and g be given by = ( ) g x- x 4. Find (f o g) ( ) x and (g o f) ( ) x . State the domain of each of the composite functions f o g and g o f. Solution : (f o g) = ( ) x ( ) f ( ) g x = ( ) f- x 4 = - x 4 , that is, (f o g) = ( ) x- x 4. _________ (g o f) = ( ) x ( ) g ( ) f x = ( ) g x = - x 4, that is, (g o f) = ( ) x- x 4. _________ In order to find the domain of f o g first note that the domain of g is the set of all real numbers ( ) ,- , and the domain of f is the set of real numbers that are greater than or equal to 0, that is, D(g) = ( ) ,- and D(f) = { x | x > 0 } = [ , ). The domain of f o g is the set of all real numbers x such that = ( ) g x- x 4 is greater than or equal to 0. Now - x 4 > 0 exactly when x > 4. Hence D(f o g) = { x | x > 4 } = [ , 4 ). ______________ The domain of g o f is the same as the domain of f, namely, D( g o f ) = [ , ). _________ Example 3 : Let the function f be given by = ( ) f x 2 x and g be given by = ( ) g x x + 1 x . Find (f o g) ( ) x and (g o f) ( ) x . State the domain of each of the composite functions f o g and g o f. Solution : (f o g) = ( ) x ( ) f ( ) g x = f x + 1 x = 2 x + 1 x = 2 ....
View Full Document

This note was uploaded on 11/12/2011 for the course MATH 111 taught by Professor Uri during the Spring '08 term at Rutgers.

Page1 / 16

composition - Composition of functions Given two functions...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online