Chabot Mathematics
§2.3 Algebra
of Functions
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot College Mathematics
1
Bruce Mayer, PE
File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp
Review §
2.2
MTH 55
 Any QUESTIONS About
• §2.2 → Function Graphs
 Any QUESTIONS About HomeWork
• §2.2 → HW-04
Chabot College Mathematics
2
Bruce Mayer, PE
File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp
Function ReVisited
 A FUNCTION is a special kind of
Correspondence between two sets. The
first set is called the Domain. The second
set is called the Range. For any member of
the domain, there is EXACTLY ONE member
of the range to which it corresponds. This
kind of correspondence is called a function
Domain
Chabot College Mathematics
3
Correspondence
Range
Bruce Mayer, PE
File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp
Function Analogy → Machinery
 The function pictured has been named f.
Here x represents an arbitrary input, and f(x)
(read “f of x,” “f at x,” or “the value of “f at x”)
represents the corresponding output.
Chabot College Mathematics
4
Bruce Mayer, PE
File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp
Implicit Domain
 If the domain of a function that is
defined by an equation is not
explicitly specified, then we take
the domain of the function to be the
LARGEST SET OF REAL
NUMBERS that result in REAL
NUMBERS AS OUTPUTS.
• i.e., DEFAULT Domain is all x’s that
produce VALID Functional RESULTS
Chabot College Mathematics
5
Bruce Mayer, PE
File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp
Example  Find Function Domain
 Find the domain of
the Function
2
f ( x) 
.
x 8
 First determine if
there is/are any
number(s) x for
which the function
cannot be
computed?”
Chabot College Mathematics
6
 Recall that an
expression is
meaningless for
Division by Zero
 So In this case the
Fcn CanNot be
computed when
x−8=0
Bruce Mayer, PE
File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp
Example  Find Function Domain
 This Fcn Undefined
for x − 8 = 0
2
f ( x) 
.
x 8
 To determine what
x-value would cause
x − 8 to be 0, we
solve the equation:
 Thus 8 is not in the
domain of f,
whereas all other
real numbers are.
 Then the domain of
f is
x | x is a Real No. and x  8
x 8  0
 x 8
Chabot College Mathematics
7
Bruce Mayer, PE
File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp
Algebra of Functions
 The Sum, Difference, Product, or
Quotient of Two Functions
 Suppose that a is in the domain of two
functions, f and g. The input a is paired
with f(a) by f and with g(a) by g.
 The outputs can then be added to
obtain: f(a) + g(a).
Chabot College Mathematics
8
Bruce Mayer, PE
File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp
Algebra of Functions

If f and g are functions and x is in the
domain of both functions, then the
“Algebra” for the two functions:
1. ( f
2. ( f
3. ( f
4. ( f
 g )( x)  f ( x)  g ( x);
 g )( x)  f ( x)  g ( x);
 g )( x)  f ( x)  g ( x);
g )( x)  f ( x) g ( x), provided g ( x)  0.
Chabot College Mathematics
9
Bruce Mayer, PE
File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp
Example  Function Algebra
 Find the Following for these Functions
f ( x)  2 x  x and g ( x)  3x  1,
2

•
a) (f + g)(4)
b) (f − g)(x)
•
c) (f/g)(x)
d) (f•g)(−1)
SOLUTION
a) Since f(4) = −8 and g(4) = 13,
we have (f + g)(4) =
f(4) + g(4) = −8 + 13 = 5.
Chabot College Mathematics
10
Bruce Mayer, PE
File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp
Example  Function Algebra
2
f
(
x
)

2
x

x
and g ( x)  3x  1,
 SOLUTION for
b) (f − g)(x) → ( f  g )( x)  f ( x)  g ( x)
 2 x  x  (3x  1)
2
  x  x  1.
c) (f/g)(x) → ( f / g )( x)  f ( x) / g ( x)
2
2x  x

.
3x  1
2
Assumes
1
x
3
Chabot College Mathematics
11
Bruce Mayer, PE
File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp
Example  Function Algebra

2
f
(
x
)

2
x

x
and g ( x)  3x  1,
SOLUTION for
d) (f•g)(−1) →
f(−1) = −3 and
g(−1) = −2, so
( f  g )(1)  f (1)  g (1)  (3)(2)  6.
Chabot College Mathematics
12
Bruce Mayer, PE
File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp
Example  Function Algebra
 Given f(x) = x2 + 2 and g(x) = x − 3,
find each of the following.
• a) The domain of f + g, f − g, f•g, and f/g
• b) (f − g)(x)
c) (f/g)(x)
 SOLUTION a)
• The domain of f is the set of all real
numbers. The domain of g is also the set of
all real numbers. The domains of f +g,
f − g, and f•g are the set of numbers in the
intersection of the domains; i.e., the set of
numbers in both domains, or all real No.s
• For f/g, we must exclude 3, since g(3) = 0
Chabot College Mathematics
13
Bruce Mayer, PE
File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp
Example  Function Algebra
 SOLUTION b) → (f − g)(x)
• (f − g)(x) = f(x) − g(x) = (x2 + 2) − (x − 3)
= x2 − x + 5
 SOLUTION c) → (f/g)(x)
f ( x)
( f / g )( x) 
g ( x)
• Remember to add the
restriction that x ≠ 3,
since 3 is not in the
domain of (f/g)(x)
Chabot College Mathematics
14
x 2

x3
2
Bruce Mayer, PE
File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp
WhiteBoard Work
 Problems From §2.3 Exercise Set
• 56 by PPT, 10, 30, 36, 42, 64

Demographers use birth
and death rates to
determine population
growth and evaluate the
general health of the
populations they study.
These rates usually denote
the number of births and
deaths per 1,000 people in
a given year.
Chabot College Mathematics
15
Bruce Mayer, PE
File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp
P2.3-56 Functions by Graphs
6
5
y  f x 
 From the Graph
the fcn T-table
y
4
3
2
x
1
0
-6
-5
-4
-3
-2
y  g x 
-1
0
-1
-2
-3
-4
-5
M55_§2.2_Graphs_0806.xls
Chabot College Mathematics
16
1
2
3
4
5
6
x
-5
-4
-3
-2
-1
0
1
2
3
4
5
f(x) g(x)
-1
5
0
4
1
3
2
3
2
2
0
1
1
-1
1
-3
-1
-2
-6
Bruce Mayer, PE
File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp
P2.3-56 Graph (f − g)(x)
 Recall from
Lecture
6
y
y1  f x 5
• (f − g)(x) =
f(x) − g(x)
4
3
 Next Use for
Plotting
• y1 = f(x)
• y2 = g(x)
2
-6
-5
-4
-3
-2
0
-1
1
2
3
4
5
-1
-2
-3
-4
• f(x) − g(x) =
y1 − y2
-5
M55_§2.2_Graphs_0806.xls
17
x
0
 Thus for Plot
Chabot College Mathematics
y2  g x 
1
-6
Bruce Mayer, PE
File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp
6
P2.3-56 Graph (f − g)(x)
 Recall from
Lecture
• (f − g)(x) =
f(x) − g(x)
 Use the above
relation to construct
(f − g)(x) T-Table
• Can only Calc
f(x) – g(x) where
Domains OverLap
Chabot College Mathematics
18
x
-5
-4
-3
-2
-1
0
1
2
3
4
5
f(x) g(x)
-1
5
0
4
1
3
2
3
2
2
0
1
1
-1
1
-3
-1
-2
f(x) - g(x)
No Domain
5
3
1
1
2
0
-2
No Domain
No Domain
No Domain
Bruce Mayer, PE
File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp
y
5
4
y   f  g x 
3
y  g x 
2
1
x
0
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
-1
-2
-3
-4
y  f x 
-5
M55_§2.2_Graphs_0806.xls
Chabot College Mathematics
19
-6
Bruce Mayer, PE
File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp
P2.3-56 (f – g)(x) Graph
6
All Done for Today
Fcn Algebra
By
MicroProcessor
Chabot College Mathematics
20
Bruce Mayer, PE
File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp
Chabot Mathematics
Appendix
r  s  r  s r  s 
2
2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
–
Chabot College Mathematics
21
Bruce Mayer, PE
File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp