COMPOSITION

OFFUNCTIONS

“SUBSTITUTING ONE FUNCTION INTO ANOTHER”

The DOMAIN of the Composition Function

The domain of f composition g is the set of all numbers x in the domain of g such that g(x) is in the domain of f.

11

xxgx

xf

1

1

xgf

The domain of g is x 1

We also have to worry about any “illegals” in this composition function, specifically dividing by 0. This would mean that x 1 so the domain of the composition would be combining the two restrictions.

1: is ofdomain xxgf

0: yy 6: xx

The DOMAIN and RANGE of Composite Functions

We could first look at the natural domain and range of f(x) and g(x).

1

15

x

xgxxf

Hence we must exclude 6 from the domain of f(x)

For g(x) to cope with the output from f(x) we must ensure that the output does not include 1

5xxf

?)( xfg

1: yy

1

1

x

xg

1 x

0: yy 6: xx

The DOMAIN and RANGE of Composite Functions

Or we could find g o f (x) and determine the domain and range of the resulting expression.

1

15

x

xgxxf

However this approach must be used with CAUTION.

6

1)(

x

xfg

Domain: Range:

5: yy 1: xx

The DOMAIN and RANGE of Composite Functions

We could first look at the natural domain and range of f(x) and g(x).

1

15

x

xgxxf

Hence we must exclude 1 from the domain of g(x)

For f(x) to cope with the output from g(x) we must ensure that the output does not include 0

1

1

x

xg

?)( xgf

0: yy

5xxf

0 x

5: yy 1: xx

The DOMAIN and RANGE of Composite Functions

Or we could find f o g (x) and determine the domain and range of the resulting expression.

1

15

x

xgxxf

However this approach must be used with CAUTION.

51

1)(

x

xgf

Domain: Range:

0: yy

0: xx

The DOMAIN and RANGE of Composite Functions

We could first look at the natural domain and range of f(x) and g(x).

2xxgxxf

xxf

?)( xfg

0: yy

2xxg

0 x

0: yy 0: xx

The DOMAIN and RANGE of Composite Functions

Or we could find g o f (x) and determine the domain and range of the resulting expression.

2xxgxxf

However this approach must be used with CAUTION.

xxfg )(

Domain: Range:

Not: yandx

0: yy 2: xx

The DOMAIN and RANGE of Composite Functions

We could first look at the natural domain and range of f(x) and g(x).

22 xxgxxf

f o g (x) is a function for the natural domain of g(x)

f(x) can cope with all the numbers in the range of g(x) because the range of g(x) is contained within the domain of f(x)

2 xxg

?)( xgf

0: yy

xxf 2

0 x

0: yy

The DOMAIN and RANGE of Composite Functions

We could first look at the natural domain and range of f(x) and g(x).

22 xxgxxf

g(x) cannot cope with all the numbers in the range of f(x). Need to restrict the domain f(x) to give an output that g(x) can cope with.

xxf 2

?)( xfg

y

2 xxg

x

0: yy 1: xx

The DOMAIN and RANGE of Composite Functions

We could first look at the natural domain and range of f(x) and g(x).

22 xxgxxfg o f (x) is not a function for the natural domain of g(x) unless we

restrict the domain of f(x)

g(x) cannot cope with all the numbers in the range of f(x). Need to restrict the domain f(x) to give an output that g(x) can cope with.

xxf 2

?)( xfg

y

2 xxg

x 2: xx 2: yy

Example of Composite Functions

14132))((5 2 xxxfgxxf

?)( xg

Try it !!

141325 2 xxxfgxxf

14132)5(

14132)(2

2

xxxg

xxxfg

Method 1

???50202

???)5(2)5(2

2

xx

xxg

1)5(7)5(2

367)5(2)5(2

2

xx

xxxg

141325 2 xxxfgxxf

Method 1

14132)5(

14132)(2

2

xxxg

xxxfg

???50202

???)5(2)5(2

2

xx

xxg

1)5(7)5(2

367)5(2)5(2

2

xx

xxxg

141325 2 xxxfgxxf

Method 1

1)5(7)5(2

367)5(2)5(2

2

xx

xxxg

172)( 2 xxxg

5 yx

141325 2 xxxfgxxf

5)( xxfyLet

Method 2

14132)(

14132)(2

2

xxyg

xxxfg

172

14651350202

14)5(13)5(2)(

2

2

2

yy

yyy

yyyg

141325 2 xxxfgxxf

Method 2

172)( 2 xxxg

172

14651350202

14)5(13)5(2)(

2

2

2

yy

yyy

yyyg