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composition

# composition - Composition of functions Given two functions...

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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 .

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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 } = [ , 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 ) = [ , 0 ). _________ 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 . + 1 x x = 2 . = + 1 x 1 + 2 x 2 that is, (f o g) = ( ) x + 2 x 2. _________ (g o f) = ( ) x ( ) g ( ) f x = g 2 x = 2 x + 1 2 x = 2 + x 2 .
The last equality follows by multiplying top and bottom of 2 x + 1 2 x by x , but there are other ways to achieve the same result. For example, = 2 x + 1 2 x 2 x + x 2 x = 2 x . = x + x 2 2 + x 2 . We should perhaps note that this simplification is not valid when = x 0. We have (g o f) = ( ) x 2 + x 2 .

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