File = MTH55_Lec-04_ec_2-2_Fcn_ 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE...

File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics§2.3 §2.3

AlgebraAlgebraof of

FunctionsFunctions



Review §Review §

Any QUESTIONS About• §2.2 → Function Graphs

Any QUESTIONS About HomeWork• §2.2 → HW-04

2.2 MTH 55



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



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.



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



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|



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).



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



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.



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



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



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



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)



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.



P2.3-56 Functions by GraphsP2.3-56 Functions by Graphs

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

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



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


Bruce Mayer, PE Chabot College Mathematics-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

-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



All Done for TodayAll Done for Today

Fcn AlgebraBy

MicroProcessor



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

AppendiAppendixx

–

srsrsr 22

Date post:	19-Jan-2018
Category:	Documents
Upload:	charla-watson
View:	219 times
Download:	1 times

File = MTH55_Lec-04_ec_2-2_Fcn_ 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE...

Documents