COMPOSITEFUNCTIONS

The composite function: fg means…

Apply the rule for g, then, apply the rule for f.

So, if f(x) = x2

and g(x) = 3x + 1

Then fg(2) = f(7) …….…since g(2) = 3(2) + 1 = 7

= 72

= 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 )

A common question….

Is fg(x) the same as gf(x)?

Well, if again we have: f(x) = x2

and g(x) = 3x + 1

As seen: fg(x) = ( 3x + 1 )2

Now gf(x) = g( x2 ) = 3x2 + 1

Which is clearly not the same as ( 3x + 1 )2

….so the answer to the question is NO !

( In general )

Example 1:

)(fg d) )hgf( c) )gh( b) )fg( a) Find

1 )h( 21 )g(

)f(Given

2

2

xxxx

xxxx

xx

a) fg(x) = f ( 1 – 2x ) = ( 1 – 2x )2

b) gh(x) =

c) hgf(x) = hg(x2) = h( 1 – 2x2 )

d) fg2(x) = fgg(x) = fg( 1 – 2x ) = f { 1 – 2( 1 – 2x ) }

= f( 4x – 1 ) = ( 4x – 1 )2

g (x1 ) = 1– 2 ( x

1 ) = x 21 –

= 11 – 2x2

Example 2: ).(f find , 1 )f(Given 17 xxx

x

We see that f n(x) = x when n is even

Example 3 Given f(x) = 2x – 1 and g(x) = x2 + x ,solve the equation gf(x) = 30.

gf(x) = g( 2x – 1 ) = ( 2x – 1 )2 + ( 2x – 1 )

= ( 4x2 – 4x + 1 ) + ( 2x – 1 ) = 4x2 – 2x

So we have: 4x2 – 2x = 30

Dividing by 2: 2x2 – x – 15 = 0

( 2x + 5 )( x – 3 ) = 0 So x = 3 or – 2.5

f ( x1 ) =x

1f(x) = ff(x) = fff(x) = x1

f(x) = f{ff(x)}=

x1and f

n (x) = when n is odd. x

1f

17(x) =

Now multiply throughout by ( 2x + 1 ):

fg(x) = Note: we have ended upwith the same value that we started with.

In this case, the function g(x) is the inverse function of f(x).

fg(x) = x – 12x + 1

f x – 12x + 1

x – 12x + 1

=

+ 1

1 – 2

Example 4: and g(x) =Given that f(x) =

x – 12x + 1

x + 11 – 2x

, find the

composite function fg(x).

( x – 1 ) + ( 2x + 1 )( 2x + 1 ) – 2( x – 1 )

3x1 + 2

=

= x

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) 3 which does not exist !

For the composite function gf(x) to exist:

Since gf(x) = g(x – 5 ) 5 x

so: x ≥ 5.the square root of a negative number is not real,

and g(x) = x

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).