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COMPOSITE FUNCTIONS

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The composite function: fg means… Apply the rule for g, then, apply the rule for f. So, if f(x) = x 2 and g(x) = 3x + 1 Then fg(2) =f(7)…….…since g(2) = 3(2) + 1 = 7 = 7 2 = 49 Alternatively, we can find the ‘rule’ for fg(x) i.e. fg(x) == ( 3x + 1 ) 2 Hence: fg(2) =( 6 + 1 ) 2 = 49 f ( 3x + 1 )

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A common question…. Is fg(x) the same as gf(x)? Well, if again we have: f(x) = x 2 and g(x) = 3x + 1 As seen: fg(x) = ( 3x + 1 ) 2 Now gf(x) =g( x 2 )= 3x 2 + 1 Which is clearly not the same as ( 3x + 1 ) 2 ….so the answer to the question is NO ! ( In general )

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Example 1: a) fg(x) =f ( 1 – 2x )= ( 1 – 2x ) 2 b) gh(x) = c) hgf(x) =hg(x 2 )= h( 1 – 2x 2 ) d) fg 2 (x) =fgg(x)= fg( 1 – 2x )= f { 1 – 2( 1 – 2x ) } = f( 4x – 1 )= ( 4x – 1 ) 2 g (g ( x 1 ) = 1– 2 ( x 1 ) = x 2 1 – = 1 1 – 2x 2

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Example 2: x We see that f n (x) = x when n is even Example 3Given f(x) = 2x – 1 and g(x) = x 2 + x, solve the equation gf(x) = 30. gf(x) =g( 2x – 1 )= ( 2x – 1 ) 2 + ( 2x – 1 ) = ( 4x 2 – 4x + 1 ) + ( 2x – 1 )= 4x 2 – 2x So we have: 4x 2 – 2x = 30 Dividing by 2:2x 2 – x – 15 = 0 ( 2x + 5 )( x – 3 ) = 0So x = 3 or – 2.5 f (f ( x 1 ) = x 1 f(x) = ff(x) =fff(x) = x 1 f(x) =f{ff(x)}= x 1 and f n (x) = when n is odd. x 1 f 17 (x) =

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Now multiply throughout by ( 2x + 1 ): fg(x) = Note: we have ended up with the same value that we started with. In this case, the function g(x) is the inverse function of f(x). fg(x) = x – 1 2x + 1 f x – 1 2x + 1 x – 1 2x + 1 = + 1 1 – 2 Example 4: and g(x) = Given that f(x) = x – 1 2x + 1 x + 1 1 – 2x, find the composite function fg(x). ( x – 1 ) + ( 2x + 1 ) ( 2x + 1 ) – 2( x – 1 ) 3x 1 + 2 = = x

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Domains Care has to be taken when considering the domain of a composite function: Consider the following: If f(x) = x – 5 Now, gf(2) = g(– 3) which does not exist ! For the composite function gf(x) to exist: Since gf(x) = g(x – 5 ) so: x ≥ 5.the square root of a negative number is not real, and g(x) =

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Summary of key points: This PowerPoint produced by R.Collins ; Updated Mar. 2010 The composite function: fg means, apply the rule for g, then, apply the rule for f. fg(x) is not the same as gf(x)…..in general. If fg(x) = x, then f(x) is the inverse of g(x) …..and g(x) is the inverse of f(x).

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