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File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp 3 Bruce Mayer, PE Chabot College Mathematics 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 DomainRange CorrespondenceTRANSCRIPT
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File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp1
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics§2.3 §2.3
AlgebraAlgebraof of
FunctionsFunctions
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File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp2
Bruce Mayer, PE Chabot College Mathematics
Review §Review §
Any QUESTIONS About• §2.2 → Function Graphs
Any QUESTIONS About HomeWork• §2.2 → HW-04
2.2 MTH 55
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File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp3
Bruce Mayer, PE Chabot College Mathematics
Function ReVisitedFunction 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 RangeCorrespondence
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File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp4
Bruce Mayer, PE Chabot College Mathematics
Function Analogy → MachineryFunction 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.
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File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp5
Bruce Mayer, PE Chabot College Mathematics
Example Example Find Function Find Function DomainDomain
Find the domain of the Function
2( ) .8
f xx
First determine if there is/are any number(s) x for which the function cannot be computed?”
Recall that an expression is meaningless for Division by Zero
So In this case the Fcn CanNot be computed when x − 8 = 0
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File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp6
Bruce Mayer, PE Chabot College Mathematics
Example Example Find Function Find Function DomainDomain
This Fcn Undefined for x − 8 = 0
2( ) .8
f xx
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
808
x
x
8 No. Real a is xxx and|
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File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp7
Bruce Mayer, PE Chabot College Mathematics
Algebra of FunctionsAlgebra 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).
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File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp8
Bruce Mayer, PE Chabot College Mathematics
Algebra of FunctionsAlgebra 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. ( )( ) ( ) ( );2. ( )( ) ( ) ( );3. ( )( ) ( ) ( );4. ( )( ) ( ) ( ), provided ( ) 0.
f g x f x g xf g x f x g xf g x f x g xf g x f x g x g x
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File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp9
Bruce Mayer, PE Chabot College Mathematics
Example Example Function Algebra Function Algebra Find the Following for these Functions
2( ) 2 and ( ) 3 1,f x x x g x x
• a) (f + g)(4) b) (f − g)(x)• c) (f/g)(x) d) (f•g)(−1)
SOLUTIONa) Since f(4) = −8 and g(4) = 13,
we have (f + g)(4) = f(4) + g(4) = −8 + 13 = 5.
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File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp10
Bruce Mayer, PE Chabot College Mathematics
Example Example Function Algebra Function Algebra
c) (f/g)(x) →
( )( ) ( ) ( )f g x f x g x 22 (3 1)x x x
2 1.x x ( / )( ) ( ) / ( )f g x f x g x
22 .3 1x xx
13
x
Assumes
2( ) 2 and ( ) 3 1,f x x x g x x
b) (f − g)(x) → SOLUTION for
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File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp11
Bruce Mayer, PE Chabot College Mathematics
Example Example Function Algebra Function Algebra2( ) 2 and ( ) 3 1,f x x x g x x SOLUTION for
( )( 1) ( 1) ( 1) ( 3)( 2) 6.f g f g
d) (f•g)(−1) → f(−1) = −3 and g(−1) = −2, so
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File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp12
Bruce Mayer, PE Chabot College Mathematics
Example Example Function Algebra 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
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File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp13
Bruce Mayer, PE Chabot College Mathematics
Example Example Function Algebra 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)
• Remember to add the restriction that x ≠ 3, since 3 is not in the domain of (f/g)(x)
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File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp14
Bruce Mayer, PE Chabot College Mathematics
WhiteBoard WorkWhiteBoard 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.
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File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp15
Bruce Mayer, PE Chabot College Mathematics
P2.3-56 Functions by GraphsP2.3-56 Functions by Graphs
-6
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1
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M55_§2.2_Graphs_0806.xls
xgy
x
y
xfy x f(x) g(x)-5 -1-4 5 0-3 4 1-2 3 2-1 3 20 2 01 1 12 -1 13 -34 -15 -2
From the Graph the fcn T-table
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File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp16
Bruce Mayer, PE Chabot College Mathematics
P2.3-56 Graph (P2.3-56 Graph (ff − − gg)()(xx)) 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
x f(x) g(x) f(x) - g(x)-5 -1 No Domain-4 5 0 5-3 4 1 3-2 3 2 1-1 3 2 10 2 0 21 1 1 02 -1 1 -23 -3 No Domain4 -1 No Domain5 -2 No Domain
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File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp17
Bruce Mayer, PE Chabot College Mathematics-6
-5
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0
1
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-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
M55_§2.2_Graphs_0806.xls
x
y P2.3-56 (P2.3-56 ( ff –
– gg )()( xx ) Graph
) Graph
xgy
xfy
xgfy
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File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp18
Bruce Mayer, PE Chabot College Mathematics
All Done for TodayAll Done for Today
Fcn AlgebraBy
MicroProcessor
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File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp19
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
AppendiAppendixx
–
srsrsr 22