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Chabot Mathematics

§2.1 Intro to
Functions
Bruce Mayer, PE
BMayer@ChabotCollege.edu

Chabot College Mathematics                                                         Bruce Mayer, PE
1                                           BMayer@ChabotCollege.edu • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt
MTH 55
Review §                 1.6

• §1.6 → Exponent Rules & Properties

• §1.6 → HW-02

Chabot College Mathematics                                                Bruce Mayer, PE
2                                  BMayer@ChabotCollege.edu • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt
Ordered 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

Chabot College Mathematics                                          Bruce Mayer, PE
3                            BMayer@ChabotCollege.edu • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt
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
Chabot College Mathematics                                          Bruce Mayer, PE
4                            BMayer@ChabotCollege.edu • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt
Mathematical 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

Chabot College Mathematics                                          Bruce Mayer, PE
5                            BMayer@ChabotCollege.edu • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt
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}
Chabot College Mathematics                                             Bruce Mayer, PE
6                               BMayer@ChabotCollege.edu • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt
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)}.
Chabot College Mathematics                                             Bruce Mayer, PE
7                               BMayer@ChabotCollege.edu • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt
FUNCTION 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.

Chabot College Mathematics                                          Bruce Mayer, PE
8                            BMayer@ChabotCollege.edu • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt
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.

Chabot College Mathematics                                          Bruce Mayer, PE
9                            BMayer@ChabotCollege.edu • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt
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.
Chabot College Mathematics                                           Bruce Mayer, PE
10                            BMayer@ChabotCollege.edu • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt
Example  Is Relation a Fcn?

Chabot College Mathematics                                          Bruce Mayer, PE
11                           BMayer@ChabotCollege.edu • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt
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)}.
Chabot College Mathematics                                                  Bruce Mayer, PE
12                                   BMayer@ChabotCollege.edu • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt
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.
Chabot College Mathematics                                            Bruce Mayer, PE
13                             BMayer@ChabotCollege.edu • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt
Example  Is Relation a Fcn?

Chabot College Mathematics                                          Bruce Mayer, PE
14                           BMayer@ChabotCollege.edu • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt
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)}.
Chabot College Mathematics                                                Bruce Mayer, PE
15                                 BMayer@ChabotCollege.edu • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt
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.
Chabot College Mathematics                                           Bruce Mayer, PE
16                            BMayer@ChabotCollege.edu • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt
Function Forms
 Functions can be described by:
• A                  x
Table              y

• A
Graph

Chabot College Mathematics                                              Bruce Mayer, PE
17                               BMayer@ChabotCollege.edu • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt
Function Forms
 Functions are MOST OFTEN
described by:
• An EQUATION
2
yx
f x   x                            2

 NOTE: f(x) ≠ “f times x”
• f(x) indicates        y  x  6x  8   2
EVALUATION of the
function AT the
INDEPENDENT           g x   x  6x  8               2
variable-value of x
Chabot College Mathematics                                               Bruce Mayer, PE
18                                BMayer@ChabotCollege.edu • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt
Evaluating a Function
 Let g be the function defined by the
equation  y = g(x) = x2 – 6x + 8
 Evaluate each function value:
 1
a. g 3       b. g 2       c. g  
 2
d. g a  2   e. g x  h 
 SOLUTION
a. g 3  3  6 3  8  1
2

Chabot College Mathematics                                              Bruce Mayer, PE
19                               BMayer@ChabotCollege.edu • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt
Evaluating a Function
 Evaluate fcn  y = g(x) = x2 – 6x + 8
 1
b. g 2       c. g  
 2
d. g a  2  e. g x  h 

 SOLUTION
b. g 2   2   6 2   8  24
2

2
 1  1      1      21
c. g       6    8 
 2  2      2      4
Chabot College Mathematics                                              Bruce Mayer, PE
20                               BMayer@ChabotCollege.edu • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt
Evaluating a Function
 Evaluate fcn  y = g(x) = x2 – 6x + 8
d. g a  2  e. g x  h 
 SOLUTION
d. g a  2   a  2   6 a  2   8
2

 a 2  4a  4  6a  12  8
 a 2  2a
e. g x  h   x  h   6 x  h   8
2

Chabot College Mathematics
 x  2xh  h  6x  6h  8
2            2
Bruce Mayer, PE
21                                        BMayer@ChabotCollege.edu • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt
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.                any value of x
6x 2  3y  12        corresponds to
6x 2  3y  3y  12  12  3y  12                                       ONE value of y
so it DOES
6x 2  12  3y                                    define y as a
function of x
2x 2  4  y
Chabot College Mathematics                                                     Bruce Mayer, PE
22                                      BMayer@ChabotCollege.edu • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt
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
y x 4
2    2
correspond to the
same value of x so
y x x 4x
2          2           2       2
the expression does
y x 4
2   2
NOT define y as a
function of x.
y x 42

Chabot College Mathematics                                                      Bruce Mayer, PE
23                                       BMayer@ChabotCollege.edu • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt
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                                           Bruce Mayer, PE
24                            BMayer@ChabotCollege.edu • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt
Example  Find the Domain
 Find the DOMAIN of each function.
1
a. f x            b. g x   x
1 x 2

1
c. h x            d. P t   2t  1
x 1
 SOLUTION
a. f is not defined when the denominator is 0.
1−x2 ≠ 0 → Domain: {x|x ≠ −1 and x ≠ 1}

Chabot College Mathematics                                               Bruce Mayer, PE
25                                BMayer@ChabotCollege.edu • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt
Example  Find the Domain
 SOLUTION b. g x   x
• The square root of a negative number is
not a real number and is thus excluded
from the domain
x NONnegative → Domain: {x|x ≥ 0}, [0, ∞)

Chabot College Mathematics                                           Bruce Mayer, PE
26                            BMayer@ChabotCollege.edu • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt
Example  Find the Domain
1
 SOLUTION c. h x  
x 1
• 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, ∞)
Chabot College Mathematics                                           Bruce Mayer, PE
27                            BMayer@ChabotCollege.edu • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt
Example  Find the Domain
 SOLUTION d. P t   2t  1
• Any real number substituted for t yields a
unique real number.

NO UNDefinition →
Domain: {t|t is a real number}, or (−∞, ∞)

Chabot College Mathematics                                            Bruce Mayer, PE
28                             BMayer@ChabotCollege.edu • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt
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.

Chabot College Mathematics                                          Bruce Mayer, PE
29                           BMayer@ChabotCollege.edu • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt
WhiteBoard Work
 Problems From
§2.1 Exercise Set                   x                          f(x)                       g(x)
• 18, 26                      -2                              6                       0
-1                              3                       4
0                            -1                        1
        P2.1-26                      1                            -4                        -3
Functional                   2                              0                       -6
Relationships

Chabot College Mathematics                                          Bruce Mayer, PE
30                           BMayer@ChabotCollege.edu • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt
All Done for Today

Some
Statin
Drugs

Chabot College Mathematics                                          Bruce Mayer, PE
31                           BMayer@ChabotCollege.edu • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt
Chabot Mathematics

Appendix
r  s  r  s r  s 
2        2

Bruce Mayer, PE
BMayer@ChabotCollege.edu

–   Chabot College Mathematics
32
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt

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