1. Functions and Their Properties

7/31/2019 1. Functions and Their Properties

1/38

FunctionsandTheirBaProperties

FunctionsandtheirGraphsTangentsandAsymptotesI j i i d S j i i


2/38

I j ti it d S j ti it

FunctionsA function is rule that assigns an element of a set, calt a r g e t d o m a in of the function, to any element of thethe function is defined. This set is the d o m a in o f d e fthe function.

Examples

The scale is a function. It assigns to every per

stepping on it, the weight of the person.1

The thermometer is a function, it can be used toutside temperature at any moment. It measurtemperature of its location continuously. The d

d fi iti i ti i t l d th t t d

2

fGraphicallyDomain ofdefinition


3/38

LengthoftheDay24

18

12

6

o

urs

HelsinkiTo understand how thelength of the day varies,one can measure thelength of the first day of

each month. That datacan be plotted.

Joining the plotted pointsby straight or slightlycurved lines yields agraph that allows one tounderstand how thelength of the day varies.The l en g t h o f t h e d ay

i f t i d th


4/38

DefinitionofFunctionsGiven sets A and B. A f u n c t i o n f : A

which assigns an element f(a) of the set B for every

If the sets A and Bare finite, then this rule can be ex

terms of a table or a diagram.Usually the sets A and Bare not finite. In such a casquestion is usually expressed in terms of an algebraic einvolving possibly special functions, for f(a).

Alternatively the rule to compute f(a), for a given a,program taking a as input and producing f(a) as its

Definition

Example ( ) ( )sinf1xx

x=

is a function which is defin


5/38

GraphsofFunctions

Examples

In calculus we are usually concerned with functions f:real number to a real number. Such functions are usuaexplicit expression for f(x).

The product 2 = {(x,y)| x, y} is called the p lane

pictured by drawing the x axis horizontally, and the yThe graph of the function f: is the graph of the set

Below are graphs of the functions f(x) = s

g(x) = x4

2x3

x2

+ 2xand h(x) = 2sin(x)

Which is which?


6/38

CurvesandGraphsProblem Which of the following curves in the plane

graphs of functions?

A Th fi t t t h f f


7/38

SecantandTangentLinesA line intersecting the graph of afunction at two points is a secantl i ne.

If you modify a secant line by

rotating it around the firstintersection point so that thesecond intersection pointapproaches the first one, you get

a t a n g en t l i n e at the limit.

A tangent lin

It h th t

No tangentat this point


8/38

AsymptotesAn asymptote of a curve is a st ra i g h t l i n ewhich the curve approaches arbitrarilyclose as one moves sufficiently far along thecurve.

Here we see the graph of the functionand its asymptotes, the line y= x,and the vertical asymptote x= 2.

1

2y xx= +

A graph may intersect its asinfinitely often. Asymptotes

i f ti b t th b h


9/38

InjectiveFunctionsDefinition

A one-to-one function associates at most one point in t

to any given point in the set B; i.e. f(x1) = f(x2) x1 =

Problem Which of the following graphs are graphs oone functions?

A function f: A B is i n j e c t i v e or o n e - t odifferent elements ofA have different imagi.e. if x1 x2 f(x1) f(x2).


10/38

SurjectiveandBijectiveFunctDefinition A function f: A B is su r jec t i ve or o n t o

covers all of the target domain; i.e. if y B: x A such that f(x) = y.

A function f: A B is b i j ec t i ve if it is both surjective i.e. if y B: ! x A such that f(x) = y.

The notation ! x A in the above means that there

element in the set A having the given property.


11/38

Increasing andDecreasingFuDefinition

A function f is inc reas ing a> b f(a)

A function, whose values grow as the value of the variacalled increasing. If the values of the function decreasvariable grows, the function is decreasing.

A function f is decreas ing a> b f(a) < f(b).

A function f is m o n o t o n i c if it is everywhere increasi

everywhere decreasing.

Theorem A monotonic function is one-to-one (inje

Proof We have to show that, if x1 x2, then also f


12/38

Summary

Functions are rules that assign an element of the targeto any element of the domain of definition of the funct

Functions are often defined by algebraic expressions li

In such a case the domain of definition is the set of nuwhich the given algebraic expression is defined.

Tangen t l i nes of the graph of a function give local infabout the function near the point of tangency. A s y m pglobal information about the graph of a function.

Functions are i n j e c t i v e, or o n e - t o -o n e , if they assignelements of the target set to different elements of the definition. Functions are b i j ec t i ve , if they are injectiv


13/38

Continuity Gently

Definition of Continuity

I di V l Th


14/38

Simplest way to define the continuity of functions is tofunction is continuous if one can draw its graph withouthe pen from the paper.

C ti F ti Di ti



15/38

Simplest way to define the continuity of functions is tofunction is continuous if one can draw its graph withouthe pen from the paper.

C i F i Di i


Continuity


16/38

Continuity

DefinitionA function f is continuous at a number a if

limxa

f(x) = f(a).


Continuity
http://find/http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


17/38

Continuity

DefinitionA function f is continuous at a number a if

limxa

f(x) = f(a).

In other words...

1 f(a) is defined;

2 the limit, limxa f(x), exists; and,

3

these two values coincide.


Continuity


18/38

Continuity

What is continuity?

Speaking loosely, a function is continuous if we can trace its graph

without having to lift our pencil from the paper. In other words, the

graph is connected.

What is the mathematical definition of continuity? How do weexpress the connectedness of a graph?

It is easier to look at some discontinuous examples to see what can

go wrong.


Continuity


19/38

Continuity

Examples

Continuity
http://find/http://goback/http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


20/38

Continuity

Examples

Continuity


21/38

Continuity

Examples

Continuity


22/38

u y

Examples


23/38

Examples of Continuous Func

1 f(x)

= x3

x

is continuous everywhere.

2Continuous for x

0, and d

at x

= 0.

3 h(x)

= sin(x)/x

is defined and continuous for

Setting h(0) = 1 extends the function h to a cfunction defined for allx.


24/38

Rules of Continuous FunctionsAssume that both functions f and g are continuous aLet c R.

Theorem The following functions are continuous atx

1 f(x) + g(x) 2 cf(x)

3 f(x) g(x) 4 f(x)/g(x) assuming tha

Proof The result follows immediately from thecorresponding properties of limits of functio

Statement of the properties of limits.Rigorous proof of the properties of limits.


25/38

Basic Continuous Functions

Since clearly the function f(x) =x

is continuous, the

Continuous Functions implies that:

1 . A ll po l ynom ia ls are continuous functions.

2 . A ll r a t i ona l f unc t i ons, i.e., all quotients R = P/Qpolynomial P and Q are continuous at all pointsfor which Q(x0

)

0.

On e ca n sh o w f u r t h e r t h a t :

1 . A ll p ow e r f u n ct i o n s xr, rReal, are continuous wh

2 . A ll f un ct i ons f ( x) = ax, a

> 0, are continuous. I

the Ex pon en t ia l Fun ct ion e x is continuous.


26/38

Examples

1 Where the function tanxis continuous?

Solution

By the previous remarks, tanx

is

continuous everywhere where it is defined.

The function tanx

= sin(x)/cos(x) is defined

for all valuesx for which cosx 0.


27/38

Examples

2 Where the function f(x) =

x

+

x

is continu

Recall that

x

= largest integer

x.


28/38

Examples

2 Where the function f(x) =

x

+

x

is continu

Recall that

x

= largest integer

x.

Solution

Observe that if n

1


29/38

Examples

3 Where the function is continuou

SolutionObserve that the numerator isdefined and continuous for x

> 0.

The denominator lnx

1 is also

defined for allx, x > 0.

The denominator takes the value0 ifx= e. At this point thefunction is undefined, and hence

e

Grapg. Tline


30/38

Examples

4 Study the continuity of the function

Solution

Since x2

and the Sine function are both continuo

composed function sin(x2

) is continuous.

1

Since 1 + sin(x2

) 0 for allx, is dealso continuous for allx.2

The numerator is defined and confor allx. The denominator x2 is continuous, and

3


31/38

Examples

5 The function is defined

Is it possible to define f(0) so that the function continuous atx

= 0?

Solution

Since

=1 + sin x2( )( ) 1

=sin x2( )

x0

1

We need to find out whether f has a limit


32/38

Examples

5 The function is defined

Is it possible to define f(0) so that the function continuous atx

= 0?

Solution(contd)

We have concluded that, if we set f(0) = function f is continuous at x

= 0.

Problems of this type can usually besolved by computing the limit (if oneexists) of the function in question at the


33/38

Tangents, Velocity, andthe Derivative

Tangents as Limits of Secant Lines

Tangent Lines

Linear Approximations of Functions

Velocity

Rates of Change


34/38

Differentiation/Introduction to Differentiation/Tangents, Velocity, and the Derivative

Tangents as Limits of Secant Lines

The basic problem that leads to differentiation is to compute theslope of a tangent line of the graph of a given function f at a given

point x0. The key observation, which allows one to compute slopes

of tangent lines is that the tangent is a certain limit of secant lines

as illustrated in the picture below.

x0 x0+h

f

A secant line intersects the graph of afunction f at two or more points.

The figure on the left shows secantlines intersecting the graph at the

points corresponding to x=x0 and

x=x0 + h.

As h approaches 0, the secant line

in question approaches the tangent

line at the point (x0,f(x0)).


35/38


Slopes of Secant Lines

The slope of a secant line intersecting the graph of a function f atpoints corresponding tox=x0 and x=x0 + h can readily be

computed using the notations defined in the picture below.

x0 x0+h

f

h

f(x0+h)

f(x0) f(x0+h)- f(x0)

As h approaches 0 (through

positive numbers), the secant

in the pictures approaches the

tangent to the graph of f atthe point (x0,f(x0)).


36/38


1

1

Tangent Lines (1)

Compute the slope of the

tangent line, at the point

(1,1), of the graph of the

function x2.

The tangent line of the graph of the function f atthe point (x0,f(x0)) is the line passing through this

point and having the slope

provided that the limit exists and is finite.

limh0

fx0+ h( ) fx

0( )

h


37/38


Tangent Lines (2)

limh0

fx0 + h( ) fx0( )h

= limh0

1 + h( )2

1

2

h.

Compute the slope of thetangent line, at the point

(1,1), of the graph of the

function x2.

limh0

1+h

2

12

h= limh0

12 +2h+h2 12h

= limh0

2h+h2

h= limh0

2+h=2.

Equation of the

tangent line is

y-1=2(x-1), i.e.,

y=2x-1.

1

1

By the definition, the slope is thelimit

This can readily be computed by expandingthe brackets:


38/38


Linear Approximations of Functions

The following pictures show, in different scales, the graph ofthe function x2 and that of its tangent line at the point (1,1).

0.77

Documents

1. Functions and Their Properties