Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

[email protected] • MTH15_Lec-01_sec_1-1_Fcn_Intro.pptx1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§1.1 Intro to

Functions



§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



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



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



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



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}



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)}.



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.



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.



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.







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)}.




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.







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)}.



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.



Function Forms Functions can be described by:

• A Table

• A Graph

yx



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



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




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




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



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



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



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



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}




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, ∞)




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




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



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



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) = ?




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




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



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




Graphically the f○g Domain Chain




Conceptually the f○g Operation Chain



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



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.



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




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




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.




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




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



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





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}





x.

Soln d.• Domain: (−∞, −1)U(−1, ∞) or {x|x ≠ −1}

d. go f x g f x g x 1 1

x 1



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.



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.



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



WhiteBoard Work

Problems From §1.1 Exercise Set• 37, 65

http://www.google.com/url?sa=i&rct=j&q=&source=images&cd=&cad=rja&docid=5HKCS_iYDHCT0M&tbnid=F4woCLrpLVepOM:&ved=0CAUQjRw&url=http://www.chegg.com/homework-help/questions-and-answers/f-x-2x2-3x-1find-difference-quotient-simplify-answer-q1487320&ei=p0bCUe6SA8SQiQLm2YGABw&bvm=bv.48175248,d.cGE&psig=AFQjCNG03cyP-klyEsmpT60Mj7b7GUsBug&ust=1371772913671311

http://www.google.com/url?sa=i&rct=j&q=apple+logo&source=images&cd=&cad=rja&docid=IZN25HMpCO4tsM&tbnid=HhYIr-Oz_5sLkM:&ved=0CAUQjRw&url=http://news.dice.com/2013/05/29/an-inside-look-at-apple/apple-logo/&ei=S0fCUcr8JeKbjAKHkICQAQ&bvm=bv.48175248,d.cGE&psig=AFQjCNHvOoE_x-QNA-WKxd1df-YapalNLA&ust=1371773129541433



All Done for Today

SomeStatinDrugs



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

Appendix

–

srsrsr 22





P1-37









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.

Documents

Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege