Download presentation

Presentation is loading. Please wait.

Published byJenna Doyle Modified over 2 years ago

1
COMPOSITE FUNCTIONS

2
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 )

3
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 )

4
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

5
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) =

6
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

7
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) =

8
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).

Similar presentations

OK

Function Composition Given f(x) = 2x + 2 and g(x) = 2, find f ºg(x). f ºg(x)=f(g(x)Start on the inside. f(g(x)) g(x) = 2, so replace it. f(g(x)) = f(2)

Function Composition Given f(x) = 2x + 2 and g(x) = 2, find f ºg(x). f ºg(x)=f(g(x)Start on the inside. f(g(x)) g(x) = 2, so replace it. f(g(x)) = f(2)

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on supply chain management of nokia Ppt on american vs british accent Ppt on surface water pollution Ppt on france in french language Ppt on art and craft movement designs Nokia acquired by microsoft ppt online Ppt on extranuclear inheritance Ppt on chapter 3 atoms and molecules youtube Ppt on principles of object-oriented programming interview questions Ppt on centring definition