Bruce Mayer, PE Registered Electrical & Mechanical Engineer BMayer@ChabotCollege
Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege
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Transcript of Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege
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Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
§1.1 Intro to
Functions
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Bruce Mayer, PE Chabot College Mathematics
§1.1 Learning Goals
Identify the domain of a function, and evaluate a function from an equation
Gain familiarity with piecewise-defined functions
Introduce and illustrate functions used in economics
Form and use composite functions in applied problems
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Bruce Mayer, PE Chabot College Mathematics
ReCall the Ordered-Pair
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; e.g.,
An ordered pair that satisfies an equation is called a solution of the eqn
16791673 16as
3,16by Satisfied7 Eqn2
?
2
xy
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Bruce Mayer, PE Chabot College Mathematics
Ordered 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
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Bruce Mayer, PE Chabot College Mathematics
Mathematical RELATION
Any SET of ordered pairs is called a relation.
The set of all first components is called the domain of the relation,
The set of all SECOND components is called the RANGE of the relation
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Bruce Mayer, PE Chabot College Mathematics
Example 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}
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Bruce Mayer, PE Chabot College Mathematics
Example 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)}.
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Bruce Mayer, PE Chabot College Mathematics
FUNCTION Defined
A function is a “Rule” which “takes” a set X to a set Y, and is a relation in which each element of X corresponds to ONE, and ONLY ONE, element of Y.
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Bruce Mayer, PE Chabot College Mathematics
Functional 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.
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Bruce Mayer, PE Chabot College Mathematics
Example 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.
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Bruce Mayer, PE Chabot College Mathematics
Example Is Relation a Fcn?
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Bruce Mayer, PE Chabot College Mathematics
Example 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)}.
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Bruce Mayer, PE Chabot College Mathematics
Example 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.
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Bruce Mayer, PE Chabot College Mathematics
Example Is Relation a Fcn?
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Example 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)}.
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Bruce Mayer, PE Chabot College Mathematics
Function 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.
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Bruce Mayer, PE Chabot College Mathematics
Function Forms Functions can be described by:
• A Table
• A Graph
yx
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Bruce Mayer, PE Chabot College Mathematics
Function Forms
Math Functions are MOST OFTEN described by:• An EQUATION
– The Eqn can be used to MAKE aTable or Graph
y x2
f x x2
y x2 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
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Bruce Mayer, PE Chabot College Mathematics
Evaluating 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
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Bruce Mayer, PE Chabot College Mathematics
Evaluating 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
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Bruce Mayer, PE Chabot College Mathematics
Evaluating 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
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Bruce Mayer, PE Chabot College Mathematics
Example 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 3y 12
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
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Bruce Mayer, PE Chabot College Mathematics
Example 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
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Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
Example 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}
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Bruce Mayer, PE Chabot College Mathematics
Example 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, ∞)
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Bruce Mayer, PE Chabot College Mathematics
Example 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
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Bruce Mayer, PE Chabot College Mathematics
Example 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
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Bruce Mayer, PE Chabot College Mathematics
Composite Functions
In the real world, functions frequently occur in which some quantity depends on a variable that, in turn, depends on yet another variable.
Functions such as these are called COMPOSITE FUNCTIONS
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Bruce Mayer, PE Chabot College Mathematics
Composing a Function
Composition with sets A & B by fcns g & f
1
3
7
A
( ) 3 1g x x
4
10
22
B
1( ) 3
2f x x
−1
2
8
C
fg
h h(x) = ?
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Bruce Mayer, PE Chabot College Mathematics
Composing a Function
From The Diagram notice that since f takes the output from g we can combine f and g to get a function h:
1
3
7
A
( ) 3 1g x x
4
10
22
B
1( ) 3
2f x x
- 1
2
8
C
h h(x) = ?
f (g (x)) = f (3x + 1)
1(3 1) 3
2x
3 5
2 2x
This Yields an eqn for h:3 5
( ) .2 2
h x x
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Bruce Mayer, PE Chabot College Mathematics
Composing a Function
The function h is the composition of f and g and is denoted f○g (read “the composition of f and g”, or “f composed with g”, or “f circle g”).
1
3
7
A
( ) 3 1g x x
4
10
22
B
1( ) 3
2f x x
- 1
2
8
C
h h(x) = ?
3 5( ) .
2 2h x x
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Bruce Mayer, PE Chabot College Mathematics
COMPOSITION OF FUNCTIONS
If f and g are two functions, the composition of function f with function g is written as f○g and is defined by the equation
f og x f g x , The function where the domain of f○g
consists of those values x in the domain of g for which g(x) is in the domain of f
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Bruce Mayer, PE Chabot College Mathematics
COMPOSITION OF FUNCTIONS
Graphically the f○g Domain Chain
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Bruce Mayer, PE Chabot College Mathematics
COMPOSITION OF FUNCTIONS
Conceptually the f○g Operation Chain
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Bruce Mayer, PE Chabot College Mathematics
Example Evaluate Composites
Given: f x x3 and g x x 1.
Find Each of the Following
a. f og 1 b. go f 1 c. f o f 1 d. gog 1
Solution a.
a. f og 1 f g 1 f 2 23
8
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Bruce Mayer, PE Chabot College Mathematics
Example Evaluate Composites
Solution b.
b. go f 1 g f 1 g 1 11 2
Solution c.
c. f o f 1 f f 1 f 1 1 3 1
Solution d.
d. gog 1 g g 1 g 0 0 1 1
f x x3 and g x x 1.
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Bruce Mayer, PE Chabot College Mathematics
Example Fcn Composition
Given f(x) = 4x and g(x) = x2 + 2, find
( ) and ( ).f g x g f x
SOLUTION
( ) ( ( ))f g x f g x = f (x2 + 2)
= 4(x2 + 2) = 4x2 + 8
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Bruce Mayer, PE Chabot College Mathematics
Example Fcn Composition
Given f(x) = 4x and g(x) = x2 + 2, find
( ) and ( ).f g x g f x SOLUTION
( ) ( ( ))g f x g f x = g(4x)
= (4x)2 + 2
= 16x2 + 2 This example shows
that in general ( ) ( ).f g x g f x
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Bruce Mayer, PE Chabot College Mathematics
Example Fcn Composition
Given: f x 2x 1 and g x x2 3. Find Each Composite Function
a. f og x b. go f x c. f o f x
a. f og x f g x f x2 3 2 x2 3 1
2x2 6 1
2x2 5
Solution a.
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Bruce Mayer, PE Chabot College Mathematics
Example Fcn Composition
Given: f x 2x 1 and g x x2 3.
Solution b. a. f og x b. go f x c. f o f x
b. go f x g f x g 2x 1 2x 1 2 3 4x2 4x 2
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Bruce Mayer, PE Chabot College Mathematics
Example Fcn Composition
Given: f x 2x 1 and g x x2 3.
Solution c. a. f og x b. go f x c. f o f x
c. f o f x f f x f 2x 1 2 2x 1 1 4x 3
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Example Composite Domain
Given: Let f x x 1 and g x 1
x.
c. Find f og x and its domain.
d. Find go f x and its domain.
b. Find go f 1 .a. Find f og 1 .
Solution a.
a. f og 1 f g 1 f 1 11 0
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Example Composite Domain
Given: Let f x x 1 and g x 1
x.
Solution b.
b. go f 1 g f 1 g 0 not defined
Soln c.
c. f og x f g x f1
x
1
x1
• Domain: (−∞, 0)U(0, ∞) or {x|x ≠ 0}
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Bruce Mayer, PE Chabot College Mathematics
Example Composite Domain
Given: Let f x x 1 and g x 1
x.
Soln d.• Domain: (−∞, −1)U(−1, ∞) or {x|x ≠ −1}
d. go f x g f x g x 1 1
x 1
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DEcomposing a Function
Given: Let H x 1
2x2 1.
Show that each of the following provides a DEcomposition of H(x)a. Express H x as f g x ,
where f x 1
x and g x 2x2 1.
b. Express H x as f g x , where f x 1
x and g x 2x2 1.
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Decomposing a Function
Solution: a. Express H x as f g x , where f x 1
x and g x 2x2 1.
a. f g x f 2x2 1
1
2x2 1
H x
Let H x 1
2x2 1.
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Bruce Mayer, PE Chabot College Mathematics
Decomposing a Function
Solution:Let H x 1
2x2 1.
b. Express H x as f g x , where f x 1
x and g x 2x2 1.
b. f g x f 2x2 1
1
2x2 1
H x
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WhiteBoard Work
Problems From §1.1 Exercise Set• 37, 65
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All Done for Today
SomeStatinDrugs
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Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
Appendix
–
srsrsr 22
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P1-37
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Function 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.