COMPOSITION OF FUNCTIONS “SUBSTITUTING ONE FUNCTION INTO ANOTHER”
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Transcript of COMPOSITION OF FUNCTIONS “SUBSTITUTING ONE FUNCTION INTO ANOTHER”
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