Lesson 1: Functions and their representations (slides)

Sec on 1.1Func ons and their Representa ons

V63.0121.001, Calculus IProfessor Ma hew Leingang

New York University

Announcements

I First WebAssign-ments are due January 31I Do the Get-to-Know-You survey for extra credit!

Section 1.1Functions and theirRepresentations

V63.0121.001, Calculus IProfessor Ma hew Leingang

New York University

Announcements

I First WebAssign-mentsare due January 31

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

ObjectivesI Understand the defini on of

func on.I Work with func onsrepresented in different ways

I Work with func ons definedpiecewise over several intervals.

I Understand and apply thedefini on of increasing anddecreasing func on.

What is a function?

Defini onA func on f is a rela on which assigns to to every element x in a setD a 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.

OutlineModeling

Examples of func onsFunc ons expressed by formulasFunc ons described numericallyFunc ons described graphicallyFunc ons described verbally

Proper es of func onsMonotonicitySymmetry

The Modeling Process

...Real-worldProblems

.. Mathema calModel

Mathema calConclusions

Real-worldPredic ons

. model.

interpret

Plato’s Cave

The Modeling Process

...Real-worldProblems

.. Mathema calModel

Mathema calConclusions

Real-worldPredic ons

. model.

interpret

s.Form

OutlineModeling

Functions expressed by formulas

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

Formula function exampleExample

Let f(x) =x+ 1x− 2

. Find the domain and range of f.

Solu on

Let f(x) =x+ 1x− 2

Solu onThe denominator is zerowhen x = 2, so the domain is all real numbersexcept 2. We write:

domain(f) = { x | x ̸= 2 }

Let f(x) =x+ 1x− 2

Solu on

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.

range(f) = { y | y ̸= 1 }

How did you get that?

start y =x+ 1x− 2

cross-mul ply 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

start y =x+ 1x− 2

cross-mul ply y(x− 2) = x+ 1

distribute xy− 2y = x+ 1collect x terms xy− x = 2y+ 1

factor x(y− 1) = 2y+ 1

start y =x+ 1x− 2

collect x terms xy− x = 2y+ 1

factor x(y− 1) = 2y+ 1

start y =x+ 1x− 2

No-no’s for expressionsI Cannot have zero in thedenominator of anexpression

I Cannot have a nega venumber under an evenroot (e.g., square root)

I Cannot have thelogarithm of a nega venumber

Piecewise-defined functionsExample

f(x) =

{x2 0 ≤ x ≤ 1;3− x 1 < x ≤ 2.

Find the domain and range of fand graph the func on.

Solu onThe domain is [0, 2]. The graphcan be drawn piecewise.

...0..

The range is [0, 2).

f(x) =

{x2 0 ≤ x ≤ 1;3− x 1 < x ≤ 2.

...0..

f(x) =

{x2 0 ≤ x ≤ 1;3− x 1 < x ≤ 2.

...0..

f(x) =

{x2 0 ≤ x ≤ 1;3− x 1 < x ≤ 2.

...0..

f(x) =

{x2 0 ≤ x ≤ 1;3− x 1 < x ≤ 2.

...0..

Functions described numerically

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

Functions defined by tables I

Example

Is this a func on? If so, what isthe range?

x f(x)1 42 53 6

Solu on

.....1 ..

.. 4..

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

Example

x f(x)1 42 53 6

Solu on

.....1 ..

.. 4..

Example

x f(x)1 42 53 6

Solu on

.....1 ..

.. 4..

Example

x f(x)1 42 53 6

Solu on

.....1 ..

.. 4..

Example

x f(x)1 42 53 6

Solu on

.....1 ..

.. 4..

Example

x f(x)1 42 53 6

Solu on

.....1 ..

.. 4..

Functions defined by tables II

Example

x f(x)1 42 43 6

Solu on

.....1 ..

.. 4..

Yes, the range is {4, 6}.

Example

x f(x)1 42 43 6

Solu on

.....1 ..

.. 4..

Example

x f(x)1 42 43 6

Solu on

.....1 ..

.. 4..

Example

x f(x)1 42 43 6

Solu on

.....1 ..

.. 4..

Example

x f(x)1 42 43 6

Solu on

.....1 ..

.. 4..

Example

x f(x)1 42 43 6

Solu on

.....1 ..

.. 4..

Functions defined by tables III

Example

x f(x)1 41 53 6

Solu on

.....1 ..

.. 4..

This is not a func on.

Example

x f(x)1 41 53 6

Solu on

.....1 ..

.. 4..

Example

x f(x)1 41 53 6

Solu on

.....1 ..

.. 4..

Example

x f(x)1 41 53 6

Solu on

.....1 ..

.. 4..

Example

x f(x)1 41 53 6

Solu on

.....1 ..

.. 4..

Example

x f(x)1 41 53 6

Solu on

.....1 ..

.. 4..

An ideal function

I Domain is the bu onsI Range is the kinds of sodathat come out

I You can press more thanone bu on to get somebrands

I But each bu on will onlygive one brand

An ideal function

I Domain is the bu ons

I Range is the kinds of sodathat come out

An ideal function

Why numerical functions matterQues on

Why use numerical func ons at all? Formula func ons are so mucheasier to work with.

Answer

I In science, func ons are o en defined by data.I Or, we observe data and assume that it’s close to some nicecon nuous func on.

Why numerical functions matterQues on

Why use numerical func ons at all? Formula func ons are so mucheasier to work with.

Answer

I In science, func ons are o en defined by data.I Or, we observe data and assume that it’s close to some nicecon nuous func on.

Numerical Function ExampleExample

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 ..

Functions described graphicallySome mes all we have is the “picture” of a func on, by which wemean, its graph.

The graph on the right represents a rela on but not a func on.

Functions described verbally

O en mes our func ons come out of nature and have verbaldescrip ons:

I The temperature T(t) in this room at me t.

I The eleva on h(θ) of the point on the equator at longitude θ.I The u lity u(x) I derive by consuming x burritos.

I The temperature T(t) in this room at me t.I The eleva on h(θ) of the point on the equator at longitude θ.

I The u lity u(x) I derive by consuming x burritos.

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

OutlineModeling

MonotonicityExample

Let P(x) be theprobability thatmy income wasat least $x lastyear. Whatmight a graph ofP(x) look like?

Solu on

..$0..

$52,115..

MonotonicityExample

Let P(x) be theprobability thatmy income wasat least $x lastyear. Whatmight a graph ofP(x) look like?

Solu on

..$0..

$52,115..

Monotonicity

Defini on

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

I A func on f is increasing if f(x1) < f(x2) whenever x1 < x2 forany two points x1 and x2 in the domain of f.

ExamplesExample

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

AnswerNot if they are all consumed at once! Strictly speaking, theinsa ability principle in economics means that u li es are alwaysincreasing func ons.

Example

Obviously, the temperature in Boise is neither increasing nordecreasing.

ExamplesExample

Example

ExamplesExample

Example

SymmetryConsider the following func ons described as words

Example

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

Example

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

What might their graphs look like?

Possible Intensity Graph

Example

Let I(x) be the intensityof light x distance froma point. Sketch apossible graph for I(x).

Solu on

y = I(x)

Possible Gravity Graph

Example

Let F(x) be thegravita onal force at apoint x distance from ablack hole. Sketch apossible graph for F(x).

Solu on

y = F(x)

Definitions

Defini on

I A func on f is called even if f(−x) = f(x) for all x in the domainof f.

I A func on f is called odd if f(−x) = −f(x) for all x in thedomain of f.

Examples

Example

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

Summary

I The fundamental unit of inves ga on in calculus is the func on.I Func ons can have many representa ons

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