Lesson 1: Functions

Section 1.1Functions and their Representations

V63.0121.021/041, Calculus I

New York University

September 8, 2010

Announcements

I First WebAssign-ments are due September 13

I First written assignment is due September 15

I Do the Get-to-Know-You survey for extra credit!

Announcements

I First WebAssign-ments aredue September 13

I First written assignment isdue September 15

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

V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 2 / 33

Function

Objectives: Functions and their Representations

I Understand the definition offunction.

I Work with functionsrepresented in different ways

I Work with functions definedpiecewise over severalintervals.

I Understand and apply thedefinition of increasing anddecreasing function.


What is a function?

Definition

A function f is a relation which assigns to to every element x in a set D asingle 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

MathematicalModel

MathematicalConclusions

Real-worldPredictions

modelsolve

interpret

test


Plato’s Cave


The Modeling Process

Real-worldProblems

MathematicalModel

MathematicalConclusions

Real-worldPredictions

modelsolve

interpret

test

Shadows Forms


Outline

Modeling




Functions expressed by formulas

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


Formula function example

Example

Let f (x) =x + 1

x − 2. Find the domain and range of f .

Solution

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

y =x + 1

x − 2=⇒ x =

2y + 1

y − 1

So as long as y 6= 1, there is an x associated to y. Therefore

domain(f ) = { x | x 6= 2 }range(f ) = { y | y 6= 1 }



Example

Let f (x) =x + 1


Solution

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

As for the range, we can solve

y =x + 1

x − 2=⇒ x =

2y + 1

y − 1





Example

Let f (x) =x + 1


Solution


y =x + 1

x − 2=⇒ x =

2y + 1

y − 1

So as long as y 6= 1, there is an x associated to y.

Therefore




Example

Let f (x) =x + 1


Solution


y =x + 1

x − 2=⇒ x =

2y + 1

y − 1




How did you get that?

start y =x + 1

x − 2

cross-multiply y(x − 2) = x + 1

distribute xy − 2y = x + 1

collect x terms xy − x = 2y + 1

factor x(y − 1) = 2y + 1

divide x =2y + 1

y − 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.

Solution

The 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 buttons

I Range is the kinds of sodathat come out

I You can press more than onebutton to get some brands

I But each button will onlygive one brand


Why numerical functions matter

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


Numerical Function Example

Here is the temperature in Boise, Idaho measured in 15-minute intervalsover 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 we mean,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 verbal descriptions:

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 for anytwo 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 nor decreasing.


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 black hole.


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 off .

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


Examples

I Even: constants, even powers, cosine

I Odd: odd powers, sine, tangent

I Neither: exp, log


Summary

I The fundamental unit of investigation in calculus is the function.

I Functions can have many representations


Lesson 1: Functions

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