- Function composition involves applying one function to the results of another function. It is written as g(f(x)) or (g o f)(x). - The domain of a composite function f o g is the set of all numbers x in the domain of g such that g(x) is in the domain of f. - To determine the domain and range of a composite function, we can either examine the individual domains and ranges of the original functions f and g, or evaluate the composite function f o g and examine its domain and range.

2. Composite Function
"Function Composition" is applying one function to the
results of another:
The result of f() is sent through g()
It is written: (g º f)(x)
Which means: g(f(x))

3. Example: f(x) = 2x+3 and g(x) = x2
"x" is just a placeholder, and to avoid confusion let's just call it "input":
f(input) = 2(input)+3
g(input) = (input)2
So, let's start:
(g º f)(x) = g(f(x))
First we apply f, then apply g to that result:
(g º f)(x) = (2x+3)2

4. What if we reverse the order of f and g?
(f º g)(x) = f(g(x))
First we apply g, then apply f to that result:
(f º g)(x) = 2x2+3

6. The sum f + g
xgxfxgf
This just says that to find the sum of two functions, add
them together. You should simplify by finding like terms.
1432
32
xxgxxf
1432
32
xxgf
424
23
xx
Combine like
terms & put in
descending
order

7. The difference f - g
xgxfxgf
To find the difference between two functions, subtract
the first from the second. CAUTION: Make sure you
distribute the – to each term of the second function. You
should simplify by combining like terms.
1432
32
xxgxxf
1432
32
xxgf
1432
32
xx
Distribute
negative
224
23
xx

8. The product f • g
xgxfxgf
To find the product of two functions, put parenthesis
around them and multiply each term from the first
function to each term of the second function.
1432
32
xxgxxf
1432
32
xxgf
31228
325
xxx
FOIL
Good idea to put in
descending order
but not required.

9. The quotient f /g
xg
xf
x
g
f
To find the quotient of two functions, put the first one
over the second.
1432
32
xxgxxf
14
32
3
2
x
x
g
f Nothing more you could do
here. (If you can reduce
these you should).

10. COMPOSITION
FUNCTIONS

11. The Composition
Function
xgfxgf 
This is read “f composition g” and means to copy the f
function down but where ever you see an x, substitute in
the g function.
1432
32
xxgxxf
3142
23
xgf 
51632321632
3636
xxxx
FOIL first and
then distribute
the 2

12. xfgxfg 
This is read “g composition f” and means to copy the g
function down but where ever you see an x, substitute in
the f function.
1432
32
xxgxxf
1324
32
xfg 
You could multiply
this out but since it’s
to the 3rd power we
won’t

13. So the first 4 operations on functions are
pretty straight forward.
The rules for the domain of functions would
apply to these combinations of functions as
well. The domain of the sum, difference or
product would be the numbers x in the
domains of both f and g.
For the quotient, you would also need to
exclude any numbers x that would make the
resulting denominator 0.

14. xffxff 
This is read “f composition f” and means to copy the f
function down but where ever you see an x, substitute in
the f function. (So sub the function into itself).
1432
32
xxgxxf
3322
22
xff 

15. 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.
1
1
xxg
x
xf
1
1
x
gf 
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:isofdomain xxgf 

16. 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
1
5
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
1x

17. 0: yy6: 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
1
5
x
xgxxf
However this approach must be used with CAUTION.
6
1
)(
x
xfg 
Domain: Range:

18. 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
1
5
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
0x

19. 5: yy1: 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
1
5
x
xgxxf
However this approach must be used with CAUTION.
5
1
1
)(
x
xgf 
Domain: Range:

20. 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).
2
xxgxxf
xxf
?)(xfg 
0: yy
2
xxg
0x

21. 0: yy0: 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.
2
xxgxxf
However this approach must be used with CAUTION.
xxfg )(
Domain: Range:
Not: yandx

22. 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
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)
2xxg
?)(xgf 
0: yy
xxf 2
0x