3.4 Composition of Functions

* Combine functions using algebraic operations.
* Create a new function by composition of functions.
* Evaluate composite functions.
* Find the domain of a composite function.
* Decompose a composite function into its component functions.

1. 3.4 Composition of Functions
Chapter 3 Functions

2. Concepts and Objectives
⚫Objectives for this section are:
⚫Combine functions using algebraic operations.
⚫Create a new function by composition of functions.
⚫Evaluate composite functions.
⚫Find the domain of a composite function.
⚫Decompose a composite function into its component
functions.

3. Operations on Functions
⚫Given two functions fand g, then for all values of xfor
which both f(x)and g(x)are defined, we can also define
the following:
⚫Sum
⚫Difference
⚫Product
⚫Quotient( )()()()fgxfxgx+=+ ( )()()()fgxfxgx−=− ()()()()fgxfxgx= ()
()
()
(), 0
fxf
xgx
ggx

=



4. Operations on Functions (cont.)
⚫Example: Let and . Find each
of the following:
a)
b)
c)
d)()
2
1fxx=+ ()35gxx=+ ( )()1fg+ ()()1 1gf=+ ()
2
51 113=+ ++ 02 18=+= ( )()3fg−− () ()
2
353 31−+

−−+= ()410 14−=−= ()()5fg ( )()
2
35551+

+= ()()02026 52== ()0
f
g


 ()
2
5
01
30
+
+
= 5
1
=

5. Operations on Functions (cont.)
⚫Example: Let and . Find
each of the following:
a)
b)
c)
d) ()89fxx=− ()21gxx=− ( )()fgx+ 8921xx=−+− ( )()fgx− 8921xx=−−− ()()fgx ( )8921xx=−− ()
f
x
g


 89
21
x
x
−
=
−

6. Operations on Functions (cont.)
⚫Example: Let and . Find
each of the following:
e) What restrictions are on the domain?
⚫There are two cases that need restrictions: taking the
square root of a negative number and dividing by zero.
⚫We address these by making sure the inside of g(x)> 0:()89fxx=− ()21gxx=− 210
21
1
2
x
x
x
−


So the domain must be 11
or ,
22
x





7. Composition of Functions
⚫If fand gare functions, then the composite function, or
composition, of gand fis defined by
⚫The domain of g∘fis the set of all numbers xin the
domain of fsuch that f(x)is in the domain of g.
⚫So, what does this mean?()() ()( )gfxgfx=

8. Composition (cont.)
⚫Example: A $40 pair of jeans is on sale for 25% off. If
you purchase the jeans before noon, the store offers an
additional 10% off. What is the final sales price of the
jeans?
We can’t just add 25% and 10% and get 35%. When
it says “additional 10%”, it means 10% off the
discounted price. So, it would be()
()
25% off: .7540$30
10% off: .9030$27
=
=

9. Evaluating Composite Functions
⚫Example: Let and .
(a) Find (b) Find ()21fxx=− ()
4
1
gx
x
=
− ()()2fg ()()3gf−

10. Evaluating Composite Functions
⚫Example: Let and .
(a) Find (b) Find
(a) ()21fxx=− ()
4
1
gx
x
=
− ()()2fg ()()3gf− ()() ()()22fgfg= ()
44
4
211
fff

===

− ()241817=−=−=

11. Evaluating Composite Functions
⚫Example: Let and .
(a) Find (b) Find
(b) ()21fxx=− ()
4
1
gx
x
=
− ()()2fg ()()3gf− ()() ()( )33gfgf−=− ()( )( )()231617ggg=−−=−−=− 441
7182
===−
−−−

12. Composites and Domains
⚫Given that and , find
(a) and its domain
The domain of fis the set of all nonnegative real
number, [0, ∞), so the domain of the composite
function is defined where g≥ 0, thus()fxx= ()42gxx=+ ()()fgx ()() ()()( )42fgxfgxfx==+ 42x=+ 420x+ 1
2
x− 1
so ,
2

−



13. Composites and Domains
⚫Given that and , find
(b) and its domain
The domain of fis the set of all nonnegative real
number, [0, ∞). Since the domain of gis the set of all
real numbers, the domain of the composite function
is also [0, ∞).()fxx= ()42gxx=+ ()()gfx ()() ()( )()gfxgfxgx== 42x=+

14. Composites and Domains (cont.)
⚫Given that and , find
and its domain()
6
3
fx
x
=
− ()
1
gx
x
= ()()fgx ()()
1
fgxf
x

=

 6
1
3
x
=
− 66
1313xx
xxx
==
−
− 6
13
x
x
=
−

15. Composites and Domains (cont.)
⚫Given that and , find
The domain of gis all real numbers except0, and the
domain of fis all real numbers except3. The expression
for g(x), therefore, cannot equal 3:()
6
3
fx
x
=
− ()
1
gx
x
= 1
3
x
= 13x= 1
3
x= ( )
11
,00,,
33

−



16. Decomposition of Functions
⚫In some cases, it is necessary to decomposea
complicated function. In other words, we can write it as
a composition of two simpler functions.
⚫Theremaybemorethanonewaytodecomposea
compositefunction,sowemaychoose the
decomposition that appears to be the most expedient.

17. Decomposition of Functions
⚫Example: Write as the composition of
two functions.
⚫We are looking for two functions, gand h, so
f(x)=g(h(x)). To do this, we look for a function inside
a function in the formula for f(x).
⚫Asonepossibility,wemightnoticethatthe
expression5‒ x
2
is inside the square root. We could
then decompose the function as ()
2
5fxx=− () ()
2
5 and hxxgxx=−= ()()( )
22
55ghxgxx=−=−

18. Classwork
⚫College Algebra2e
⚫3.4: 6-16 (even); 3.3: 16-24 (even); 3.2: 38-54 (even)
⚫3.4 Classwork Check
⚫Quiz 3.3