Week1 Functions

7/30/2019 Week1 Functions

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FUNCTIONS
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Learning Outcome

Understand the conceptsof function

Learning outcome

Know the properties offunctions, operation on

functions, and graphs of

functions
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3

Introduction

are the major tools for describing the real world in mathematical terms

Functions

Eg . The amount of

interest, Iwill depends on

the interest rate, rofferedby the bank

The values of one variable often depend on the values for another

)(:

"ofequals""offunctionais"

)(

rfIEg

xfyxy

xfy

Dependent variable

Independent variable

Notation

Functions is used to describe the dependence of one quantity on another
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4

Introduction

The area A of a circle depends on its radius r by the equation :

2

)( rrfA

Independent

variable

Dependent

variableSo we say that,

A is a function

of r

Examples


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5

Introduction

The velocity v of a ball falling freely in the

Earths gravitational field increases with time,until it hits the ground

So we say that, v

is a function of t

In a bacteria culture , the number n of bacteria

present after 1 hour of growth depends on thenumber no of bacteria present Initially

So we say that,

n is a function no

Examples


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6

Introduction

DefinitionDef 1.1.1.

If a variable y depends on a variable x in a such a way that each

value of x determines exactly one value of y, then we say that

y is a function of x

Def 1.1.2.

A function f is a rule that associates a unique output with each

input rangedomain

A function ( )

rangedomain

Not a function ()
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Properties

1. Independent & dependent variables

2. Domain & range

)(xfy

Ind ependent var iable =

domain/ input

Dependent variable

=image/range/output

If the independent and dependent variables are real numbers, we say

that fis a real valued fun ct ion of a real var iable
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Properties

Function Domain Range

All possible inputs Set of outputs2xy

2,2 xxy

x

x2

y

y4

So to fully describe a function

you must not only specify the

rule that relates the inputs and

outputs, but you must also

specify the domain


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9

Properties

3. Graphs of function

Graphs can provide useful visual information about a function

The values of x for which f(x)=0 are the coordinates of the pointswhere the graph of f intersects the x-axis; These value of x are calledthe zeros of f, the roots of f(x)=0, or th e x-interc epts o f y=f(x)


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Properties

4. The vertical line test

A curve in the xy-plane is the graph of some function f if and

only if no vertical line intersects the curve more than once.

Not a function ()2522 yx
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Operations

Two functions, f and g, can be added, substracted, multiplied, and

divided in a natural way to form a new functions f+g, f-g, fg, f/g.

Def. 1.4.1 Given functions f and g, we define

)()())(( xgxfxgf )()())(( xgxfxgf

)()())(( xgxfxfg

)(/)())(/( xgxfxgf For the function f+g, f-g, and fg, we define the domain to be the

intersection of the domains of f and g, and the function f/g we

define the domain to be the intersection of the domains of f and

g but with the points where g(x)=0 excluded

1. Arithmetic operations
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Operations

2. Composition of function

Is performed by substituting some function for the

independent variable of another function

Eg.: Suppose that

2)( xxf and 1)( xxg

))(( xgf2))(( xg

2)1( x gfDenote as,


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13

Operations

3. Symmetry

about the y-axis about the x-axis about the origin

If replacing x byxin its equationproduces an

equivalent equation

If replacing y byyin its equationproduces an

equivalent equation

If replacing both x byx and y byy in itsequation produces

an equivalent

equation


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14

Operations

4. Even & odd function

Even function Odd function

)()( xfxf )()( xfxf

Examples:

x

xxx

cos&

,, 642

xs

xxx

in&

,, 53


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15

Operations

6. Translation

Operation on

y=f(x)

Add a

positive c to

f(x)

subtract a

positive c

from f(x)

Original eqn.

Geommetric

effect

Shifts up c

units

Shifts down c

units

Example

cxfy )( cxfy )(

2

-2

2xy 22 xy 22 xy
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Graphs(Families of Functions)

1. Family of linescxf )(

A function f whose value are all the same is

called a constant function

The graph of the constant function

is the horizontal linecxf )(

cy

If we vary c, then we obtain a set or family of

horizontal lines

cmxy
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cmxy Fixed c vary m

Family of lines whose

members all have y-intercept b

Fixed m, vary c

Family of parallel lines whose

members all have slope m

x

y

x

y


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n

xy 2. Power Function1n

4n

3n2n

5n


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3. Equilateral Hyperbolan

xy

1n 2n


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n

xy

1

4.

2n


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BxAy sin BxAy cos5. &

xy sin xy sin2


22/24

22


xy cos xy cos2


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xy 2sin

))2(2sin(

xy

)cos( CBxAy )sin( CBxAy 6.


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xy 2cos

))2(2cos( xy
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Week1 Functions

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Transcript of Week1 Functions