[email protected] mth55_lec-04_sec_2-1_fcn_intro.ppt 1 bruce mayer, pe chabot college...
TRANSCRIPT
[email protected] • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt1
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
§2.1 Intro §2.1 Intro toto
FunctionsFunctions
[email protected] • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt2
Bruce Mayer, PE Chabot College Mathematics
Review §Review §
Any QUESTIONS About• §1.6 → Exponent Rules & Properties
Any QUESTIONS About HomeWork• §1.6 → HW-02
1.6 MTH 55
[email protected] • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt3
Bruce Mayer, PE Chabot College Mathematics
Ordered Pair DefinedOrdered Pair Defined
An ordered pair (a, b) is said to satisfy an equation with variables a and b if, when a is substituted for x and b is substituted for y in the equation, the resulting statement is true.
An ordered pair that satisfies an equation is called a solution of the equation
[email protected] • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt4
Bruce Mayer, PE Chabot College Mathematics
Ordered Pair DependencyOrdered Pair Dependency
Frequently, the numerical values of the variable y can be determined by assigning appropriate values to the variable x. For this reason, y is sometimes referred to as the dependent variable and x as the independent variable.• i.e., if we KNOW x,
we can CALCULATE y
[email protected] • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt5
Bruce Mayer, PE Chabot College Mathematics
Mathematical RELATIONMathematical RELATION
Any set of ordered pairs is called a relation. The set of all first components is called the domain of the relation, and the set of all SECOND components is called the RANGE of the relation
[email protected] • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt6
Bruce Mayer, PE Chabot College Mathematics
Example Example Domain & Range Domain & Range
Find the Domain and Range of the relation:• { (Titanic, $600.8), (Star Wars IV, $461.0),
(Shrek 2, $441.2), (E.T., $435.1), (Star Wars I, $431.1), (Spider-Man, $403.7)}
SOLUTION• The DOMAIN is the set of all first
components, or {Titanic, Star Wars IV, Shrek 2, E.T., Star Wars I, Spider-Man}
[email protected] • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt7
Bruce Mayer, PE Chabot College Mathematics
Example Example Domain & Range Domain & Range
Find the Domain and Range for the relation:• { (Titanic, $600.8), (Star Wars IV, $461.0),
(Shrek 2, $441.2), (E.T., $435.1), (Star Wars I, $431.1), (Spider-Man, $403.7)}
SOLUTION• The RANGE is the set of all
second components, or {$600.8, $461.0, $441.2, $435.1, $431.1, $403.7)}.
[email protected] • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt8
Bruce Mayer, PE Chabot College Mathematics
FUNCTION DefinedFUNCTION Defined
A function which “takes” a set X to a set Y is a relation in which each element of X corresponds to ONE, and ONLY ONE, element of Y.
[email protected] • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt9
Bruce Mayer, PE Chabot College Mathematics
Functional CorrespondenceFunctional Correspondence A relation may be defined by a
correspondence diagram, in which an arrow points from each domain element to the element or elements in the range that correspond to it.
[email protected] • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt10
Bruce Mayer, PE Chabot College Mathematics
Example Example Is Relation a Fcn? Is Relation a Fcn?
Determine whether the relations that follow are functions. The domain of each relation is the family consisting of Malcolm (father), Maria (mother), Ellen (daughter), and Duane (son).
1. For the relation defined by the following diagram, the range consists of the ages of the four family members, and each family member corresponds to that family member’s age.
[email protected] • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt11
Bruce Mayer, PE Chabot College Mathematics
Example Example Is Relation a Fcn? Is Relation a Fcn?
[email protected] • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt12
Bruce Mayer, PE Chabot College Mathematics
Example Example Is Relation a Fcn? Is Relation a Fcn?
1. SOLUTION: The relation IS a FUNCTION, because each element in the domain corresponds to exactly ONE element in the range.
• For a function, it IS permissible for the same range element to correspond to different domain elements. The set of ordered pairs that define this relation is {(Malcolm, 36), (Maria, 32), (Ellen, 11), (Duane, 11)}.
[email protected] • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt13
Bruce Mayer, PE Chabot College Mathematics
Example Example Is Relation a Fcn? Is Relation a Fcn?
2. For the relation defined by the diagram on the next slide, the range consists of the family’s home phone number, the office phone numbers for both Malcolm and Maria, and the cell phone number for Maria. Each family member corresponds to all phone numbers at which that family member can be reached.
[email protected] • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt14
Bruce Mayer, PE Chabot College Mathematics
Example Example Is Relation a Fcn? Is Relation a Fcn?
[email protected] • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt15
Bruce Mayer, PE Chabot College Mathematics
Example Example Is Relation a Fcn? Is Relation a Fcn?
2. SOLUTION: The relation is NOT a function, because more than one range element corresponds to the same domain element. For example, both an office ph. number and a home ph. number correspond to Malcolm.
• The set of ordered pairs that define this relation is {(Malcolm, 220-307-4112), (Malcolm, 220-527-6277 ), (MARIA, 220-527-6277), (MARIA, 220-416-5204), (MARIA, 220-433-8195), (Ellen, 220-527-6277), (Duane, 220-527-6277)}.
[email protected] • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt16
Bruce Mayer, PE Chabot College Mathematics
Function NotationFunction Notation Typically use single letters such as f, F, g, G,
h, H, and so on as the name of a function. For each x in the domain of f, there
corresponds a unique y in its range. The number y is denoted by f(x) read as “f of x” or “f at x”.
We call f(x) the value of f at the number x and say that f assigns the f(x) value to y. • Since the value of y depends on the given value
of x, y is called the dependent variable and x is called the independent variable.
[email protected] • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt17
Bruce Mayer, PE Chabot College Mathematics
Function FormsFunction Forms Functions can be described by:
• A Table
• A Graph
yx
[email protected] • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt18
Bruce Mayer, PE Chabot College Mathematics
Function FormsFunction Forms
Functions are MOST OFTEN described by:• An EQUATION yx2
f x x2
yx2 6x 8
g x x2 6x 8
NOTE: f(x) ≠ “f times x”• f(x) indicates
EVALUATION of the function AT the INDEPENDENT variable-value of x
[email protected] • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt19
Bruce Mayer, PE Chabot College Mathematics
Evaluating a FunctionEvaluating a Function
Let g be the function defined by the equation y = g(x) = x2 – 6x + 8
Evaluate each function value:
a. g 3 b. g 2 c. g1
2
d. g a 2 e. g x h SOLUTION
a. g 3 32 6 3 8 1
[email protected] • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt20
Bruce Mayer, PE Chabot College Mathematics
Evaluating a FunctionEvaluating a Function
Evaluate fcn y = g(x) = x2 – 6x + 8
b. g 2 c. g1
2
d. g a 2 e. g x h SOLUTION
b. g 2 2 2 6 2 8 24
c. g1
2
1
2
2
61
2
8
21
4
[email protected] • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt21
Bruce Mayer, PE Chabot College Mathematics
Evaluating a FunctionEvaluating a Function
Evaluate fcn y = g(x) = x2 – 6x + 8
d. g a 2 e. g x h SOLUTIONd. g a 2 a 2 2 6 a 2 8
a2 4a 4 6a 12 8
a2 2a
e. g x h x h 2 6 x h 8
x2 2xh h2 6x 6h 8
[email protected] • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt22
Bruce Mayer, PE Chabot College Mathematics
Example Example is an EQN a FCN?? is an EQN a FCN??
Determine whether each equation determines y as a function of x.
a. 6x2 – 3y = 12 b. y2 – x2 = 4 SOLUTION a.
6x2 3y12
6x2 3y 3y 12 12 3y 12
6x2 12 3y
2x2 4 y
any value of x corresponds to ONE value of y so it DOES define y as a function of x
[email protected] • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt23
Bruce Mayer, PE Chabot College Mathematics
Example Example is an EQN a FCN?? is an EQN a FCN??
Determine whether each equation determines y as a function of x.
a. 6x2 – 3y = 12 b. y2 – x2 = 4 SOLUTION b. TWO values of y
correspond to the same value of x so the expression does NOT define y as a function of x.
y2 x2 4
y2 x2 x2 4 x2
y2 x2 4
y x2 4
[email protected] • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt24
Bruce Mayer, PE Chabot College Mathematics
Implicit DomainImplicit 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
[email protected] • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt25
Bruce Mayer, PE Chabot College Mathematics
Example Example Find the Domain Find the Domain
Find the DOMAIN of each function.
a. f x 1
1 x2 b. g x x
c. h x 1
x 1d. P t 2t 1
SOLUTIONa. f is not defined when the denominator is 0.
1−x2 ≠ 0 → Domain: {x|x ≠ −1 and x ≠ 1}
[email protected] • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt26
Bruce Mayer, PE Chabot College Mathematics
Example Example Find the Domain Find the Domain
SOLUTION
• The square root of a negative number is not a real number and is thus excluded from the domain
b. g x x
x NONnegative → Domain: {x|x ≥ 0}, [0, ∞)
[email protected] • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt27
Bruce Mayer, PE Chabot College Mathematics
Example Example Find the Domain Find the Domain
SOLUTION
• The square root of a negative number is not a real number and is excluded from the domain, so x − 1 ≥ 0. Thus have x ≥ 1
• However, the denominator must ≠ 0, and it does = 0 when x = 1. So x = 1 must be excluded from the domain as well
DeNom NONnegative-&-NONzero → Domain: {x|x > 1}, (1, ∞)
c. h x 1
x 1
[email protected] • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt28
Bruce Mayer, PE Chabot College Mathematics
Example Example Find the Domain Find the Domain
SOLUTION
• Any real number substituted for t yields a unique real number.
NO UNDefinition → Domain: {t|t is a real number}, or (−∞, ∞)
d. P t 2t 1
[email protected] • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt29
Bruce Mayer, PE Chabot College Mathematics
Function EqualityFunction Equality
Two functions f and g are equal if and only if:
1. f and g have the same domain • and
2. f(x) = g(x) for all x in the domain.
[email protected] • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt30
Bruce Mayer, PE Chabot College Mathematics
WhiteBoard WorkWhiteBoard Work
Problems From §2.1 Exercise Set• 18, 26
P2.1-26 FunctionalRelationships
x f(x) g(x)
-2 6 0
-1 3 4
0 -1 1
1 -4 -3
2 0 -6
[email protected] • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt31
Bruce Mayer, PE Chabot College Mathematics
All Done for TodayAll Done for Today
SomeStatinDrugs
[email protected] • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt32
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
AppendiAppendixx
–
srsrsr 22