Lesson 1: Functions

Section 1.1Functions and their Representations

V63.0121.021/041, Calculus I

New York University

September 7, 2010

Announcements

I First WebAssign-ments are due September 14I Do the Get-to-Know-You survey for extra credit!

. . . . . .

. . . . . .

Announcements

I First WebAssign-ments aredue September 14

I Do the Get-to-Know-Yousurvey for extra credit!

V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 2 / 33

Function

. . . . . .

. . . . . .

Objectives: Functions and their Representations

I Understand the definitionof function.

I Work with functionsrepresented in differentways

I Work with functionsdefined piecewise overseveral intervals.

I Understand and apply thedefinition of increasing anddecreasing function.


. . . . . .

What is a function?

DefinitionA function f is a relation which assigns to to every element x in a set Da single element f(x) in a set E.

I The set D is called the domain of f.I The set E is called the target of f.I The set { y | y = f(x) for some x } is called the range of f.


. . . . . .

Outline

Modeling

Examples of functionsFunctions expressed by formulasFunctions described numericallyFunctions described graphicallyFunctions described verbally

Properties of functionsMonotonicitySymmetry


. . . . . .

The Modeling Process

...Real-worldProblems

..Mathematical

Model

..MathematicalConclusions

..Real-worldPredictions

.model.solve

.interpret

.test


. . . . . .

Plato's Cave


. . . . . .

The Modeling Process

...Real-worldProblems

..Mathematical

Model

..MathematicalConclusions

..Real-worldPredictions

.model.solve

.interpret

.test

.Shadows .Forms


. . . . . .

Outline

Modeling




. . . . . .

Functions expressed by formulas

Any expression in a single variable x defines a function. In this case,the domain is understood to be the largest set of x which aftersubstitution, give a real number.


. . . . . .

Formula function example

Example

Let f(x) =x+ 1x− 2

. Find the domain and range of f.

SolutionThe denominator is zero when x = 2, so the domain is all real numbersexcept 2. As for the range, we can solve

y =x+ 1x− 2

=⇒ x =2y+ 1y− 1

So as long as y ̸= 1, there is an x associated to y. Therefore

domain(f) = { x | x ̸= 2 }range(f) = { y | y ̸= 1 }


. . . . . .


Example

Let f(x) =x+ 1x− 2


SolutionThe denominator is zero when x = 2, so the domain is all real numbersexcept 2.

As for the range, we can solve

y =x+ 1x− 2

=⇒ x =2y+ 1y− 1




. . . . . .


Example

Let f(x) =x+ 1x− 2



y =x+ 1x− 2

=⇒ x =2y+ 1y− 1

So as long as y ̸= 1, there is an x associated to y.

Therefore



. . . . . .


Example

Let f(x) =x+ 1x− 2



y =x+ 1x− 2

=⇒ x =2y+ 1y− 1




. . . . . .

How did you get that?

start y =x+ 1x− 2

cross-multiply y(x− 2) = x+ 1distribute xy− 2y = x+ 1

collect x terms xy− x = 2y+ 1factor x(y− 1) = 2y+ 1

divide x =2y+ 1y− 1


. . . . . .

No-no's for expressions

I Cannot have zero in thedenominator of anexpression

I Cannot have a negativenumber under an even root(e.g., square root)

I Cannot have the logarithmof a negative number


. . . . . .

Piecewise-defined functions

Example

Let

f(x) =

{x2 0 ≤ x ≤ 1;3− x 1 < x ≤ 2.

Find the domain and range of f and graph the function.

SolutionThe domain is [0,2]. The range is [0,2). The graph is piecewise.

...0

..1

..2

..1

..2

.

.

.


. . . . . .

Functions described numerically

We can just describe a function by a table of values, or a diagram.


. . . . . .

Example

Is this a function? If so, what is the range?

x f(x)1 42 53 6

.

. .

..1

..2

..3

. .4

. .5

. .6

Yes, the range is {4,5,6}.


. . . . . .

Example

Is this a function? If so, what is the range?

x f(x)1 42 43 6

.

. .

..1

..2

..3

. .4

. .5

. .6

Yes, the range is {4,6}.


. . . . . .

Example

How about this one?

x f(x)1 41 53 6

.

. .

..1

..2

..3

. .4

. .5

. .6

No, that one’s not “deterministic.”


. . . . . .

An ideal function

I Domain is the buttonsI Range is the kinds of soda

that come outI You can press more than

one button to get somebrands

I But each button will onlygive one brand


. . . . . .

An ideal function

I Domain is the buttons

I Range is the kinds of sodathat come out

I You can press more thanone button to get somebrands



. . . . . .

An ideal function


that come out

I You can press more thanone button to get somebrands



. . . . . .

An ideal function


that come outI You can press more than

one button to get somebrands



. . . . . .

Why numerical functions matter

In science, functions are often defined by data. Or, we observe dataand assume that it’s close to some nice continuous function.


. . . . . .

Numerical Function Example

Here is the temperature in Boise, Idaho measured in 15-minuteintervals over the period August 22–29, 2008.

...8/22

..8/23

..8/24

..8/25

..8/26

..8/27

..8/28

..8/29

..10

..20

..30

..40

..50

..60

..70

..80

..90

..100


. . . . . .

Functions described graphically

Sometimes all we have is the “picture” of a function, by which wemean, its graph.

.

.

The one on the right is a relation but not a function.


. . . . . .

Functions described verbally

Oftentimes our functions come out of nature and have verbaldescriptions:

I The temperature T(t) in this room at time t.I The elevation h(θ) of the point on the equator at longitude θ.I The utility u(x) I derive by consuming x burritos.


. . . . . .

Outline

Modeling




. . . . . .

Monotonicity

Example

Let P(x) be the probability that my income was at least $x last year.What might a graph of P(x) look like?

.

..1

..0.5

..$0

..$52,115

..$100K


. . . . . .

Monotonicity

Definition

I A function f is decreasing if f(x1) > f(x2) whenever x1 < x2 forany two points x1 and x2 in the domain of f.

I A function f is increasing if f(x1) < f(x2) whenever x1 < x2 for anytwo points x1 and x2 in the domain of f.


. . . . . .

Examples

Example

Going back to the burrito function, would you call it increasing?

Example

Obviously, the temperature in Boise is neither increasing nordecreasing.


. . . . . .

Symmetry

Example

Let I(x) be the intensity of light x distance from a point.

Example

Let F(x) be the gravitational force at a point x distance from a blackhole.


. . . . . .

Possible Intensity Graph

..x

.y = I(x)


. . . . . .

Possible Gravity Graph

..x

.y = F(x)


. . . . . .

Definitions

Definition

I A function f is called even if f(−x) = f(x) for all x in the domain of f.I A function f is called odd if f(−x) = −f(x) for all x in the domain of

f.


. . . . . .

Examples

I Even: constants, even powers, cosineI Odd: odd powers, sine, tangentI Neither: exp, log


. . . . . .

Summary

I The fundamental unit of investigation in calculus is the function.I Functions can have many representations


Lesson 1: Functions

Documents

Transcript of Lesson 1: Functions