2.8 Function Operations and Composition

Arithmetic operations on functions
The Difference Quotient
Composition of Functions

1. 2.8 Function Operations
Chapter 2 Graphs and Functions

2. Concepts and Objectives
Function operations
Arithmetic operations on functions
The Difference Quotient
Composition of Functions and Domain

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. The Difference Quotient
Suppose that point Plies on the graph of ,yfx
•Qx+h,fx+h
h
•
Px, fx
0
y= fx
Secant line
and suppose his a
positive number.

8. The Difference Quotient
Suppose that point Plies on the graph of ,yfx
•Qx+h,fx+h
h
•
Px, fx
0
y= fx
Secant line
and suppose his a
positive number. 

fxhfx
m
xhx



With these coordinates,
the slope of the line
joining Pand Qis

9. The Difference Quotient
Suppose that point Plies on the graph of ,yfx
•Qx+h,fx+h
h
•
Px, fx
0
y= fx
Secant line
and suppose his a
positive number. 

fxhfx
m
xhx


 fxhfx
h


With these coordinates,
the slope of the line
joining Pand Qis

10. The Difference Quotient
Suppose that point Plies on the graph of ,yfx
•Qx+h,fx+h
h
•
Px, fx
0
y= fx
Secant line
and suppose his a
positive number. 

fxhfx
m
xhx


 fxhfx
h


This slope is called the
difference quotientand the
line is called a secant line.
With these coordinates,
the slope of the line
joining Pand Qis

11. The Difference Quotient (cont.)
Example: Let . Find the difference
quotient and simplify the expression.
2
23fxxx

12. The Difference Quotient (cont.)
Example: Let . Find the difference
quotient and simplify the expression.
There are three pieces of the difference quotient:
fx+h, fx, and h. We already have fxand h, so we
just have to figure out fx+h:
2
23fxxx 
2
23fxhxhxh

13. The Difference Quotient (cont.)
Example: Let . Find the difference
quotient and simplify the expression.
There are three pieces of the difference quotient:
fx+h, fx, and h. We already have fxand h, so we
just have to figure out fx+h:
2
23fxxx 
2
23fxhxhxh  
22
223xxhhxh 22
24233xxhhxh

14. The Difference Quotient (cont.)
Example (cont.): Now we put everything together.  
2 2 2
2423323xxhhxhxxfxhfx
hh

 2 2 2
2423323xxhhxhxx
h

 2
423xhhh
h

  423
423
hxh
xh
h



15. 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

16. 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



17. Evaluating Composite Functions
Example: Let and .
(a) Find (b) Find 21fxx 
4
1
gx
x

 2fg 3gf

18. Evaluating Composite Functions
Example: Let and .
(a) Find (b) Find
(a) 21fxx 
4
1
gx
x

 2fg 3gf  22fgfg 
44
4
211
fff



 241817

19. Evaluating Composite Functions
Example: Let and .
(a) Find (b) Find
(b) 21fxx 
4
1
gx
x

 2fg 3gf   33gfgf   231617ggg 441
7182



20. 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





21. 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

22. 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



23. 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





24. Classwork
2.8 Assignment (College Algebra)
2.8 –pg. 282: 2-14 (even); 2.7 –pg. 271: 24-36
(even); 2.6 –pg. 257: 48-52, 56 (even)
Classwork Check 2.8
Quiz 2.7