functions - fcampena...prepared by: ms sonia tan domain = the set of all feasible values of x for...

FUNCTIONS

Prepared by: Ms Sonia Tan

1 a

b2

3

X Y

c

A function is a pairing that assigns to

each element of the set X exactly one

element of the set Y.

FUNCTION

Using Diagrams

Function Not a Function(only a relation)

1a

b

4

X Y

c

2

3

One to One Many to One One to Many

1 a

b

4

X Y

c

2

3d


Using Ordered pairs

A relation is any set of ordered pairs.

A function is a set of ordered pairs such that no two

distinct ordered pairs have the same first coordinate.

{(1, a), (2, b), (3, d), (4, c)} {(1, a), (1, b), (2, b), (3, c)}

REAL LIFE EXAMPLES

State whether a function or not.

1. A student in a math class paired with his ID number.

2. A person paired with the type of car he or she has.

{(Annette, 123), (Judy, 049), (Sonia, 254)}

{ (Howard, Toyota), (Howard, BMW), (Janet, Honda)}

{(1, a), (2, a), (3, b), (4, b)}


2y 2x x 5

Using Tables

Time 2006 2007 2008 2009

Population 5000 6800 9500 11400

An equation of the form y = f(x)

where each value of x , we get exactly one value of y.

Using Equations

Examples:

y = 2x + 4


2y x

1 2

x

y

x

y

Using Graphs

x -2 -1 0 1

y

x 0 1 4

y 0

A Function

y = 2x + 4

Not a Function

•

•

•

•••

•

•

•

0 2 4 6

Graph of a function f is the set of all points (x,y) on the plane that

satisfy the equation y= f(x) .


Vertical Line Test for Functions

If any vertical line intersects the graph of an

equation at exactly one point, then the equation

defines y as a function of x.

x

y

x

y



21. y x

22. y x 1

Exercises: State whether a function or not

using graphs

1xy.3 xy.43

21. y x 2

2. y x 1

function

x -2 -1 0 1 2

y 4 1 0 1 4

x -1 0 1 2 3

y 0 1 2 3 2

not a function

relationPrepared by: Ms Sonia Tan

function

x -1 0 1 2 2

y 0 -1 -2

x -8 -1 0 1 8

y -2 -1 0 1 2

1xy.3

2 3

31

xyxy.43

function


FUNCTION VALUES

Given functions f(x) , f (a) means the value of

the function f at x = a. It is read as “f of a”

3x2x)x(f.Ex2

)2(f)h(f )2x(f


Evaluate following:

Sum: f+g (x) = f(x) + g(x)

OPERATIONS ON FUNCTIONS

Given functions f(x) and g(x), we can have the

Difference: f-g (x) = f(x) - g(x)

Product: )x(g)x(f)x(gf

Quotient: )x(g

)x(f)x(

g

f

Composition of functions:

))x(g(f)x(gf ))x(f(g)x(fg


EXAMPLE: Given f(x) = 2x – 5

g(x) = x2 + x – 6

Domain = the set

of all first

elements of the

ordered pairs

Example: {( -1, 1), ( 0, 0), ( 2, 4), ( 4, 16)}

Range = the set

of all second

elements of the

ordered pairs

DOMAIN AND RANGE

Give the domain and range of

{(1, 5), (3, 8), (5, 11), (7, 8), (9, -4)}

D = { -1, 0, 2, 4 } R = { 1, 0, 4, 16 }


Domain = the set

of all feasible

values of x for

which y is defined

Range = the set of

all possible resulting

values of y or images

of x under the f

For mathematical equation y = f(x)

y = f(x) = – 3x +4

Df : R

Rf : R

y

•

x•


y = h(x) =

Df : all x R such that

x – 2 0 or x 2

Rf : y 0

2x


y = h(x) =


Rf : y 0

4x2

2xor2x2x

4x4x04x222


y = h(x) =


Rf : y 0

2x4

2x22x4x

4x4x0x4

2

222


Examples: f(x) = 2x + 5 a = 2 , b = 5

f(x) = -4x a = -4 , b = 0

f(x) = 10 a = 0 , b = 10

LINEAR FUNCTIONS

f(x) = ax + b where a and b are real numbers

The graph of a linear function is a straight line.


Df : R

Rf : R

QUADRATIC FUNCTIONS

f(x) = ax2 + bx + c where a, b and c are real numbers

and .

The graph of quadratic function :

0a

V(h,k)

If a > 0, parabola upward. If a < 0, parabola downward.

V(h,k)


f(x) = ax2 + bx + c

V = (h, k) is called the vertex of the parabola.

V(h,k)

V(h,k)

f(x) = a(x - h)2 + k

a2

bh

where and k = f(h)

x = h

x = hLowest point

Highest point

R:Dfky:R f ky:R f R:Df


k = f(-1) = (-1)2 + 2(-1) -8 = -9

V = (-1, -9) f(x) = (x + 1)2 - 9

12

2

a2

bh

Examples: f(x) = x2 + 2x - 8 a = 1 , b = 2 , c = -8

V = (-1, -9)•

R:Df 9y:R f Prepared by: Ms Sonia Tan

Examples: f(x) = -x2 + x +2 a = -1 , b = 1 , c = 2

•

),(V4

9

21

4

92

21 )x()x(f

),(V4

9

21

4

9f y:R R:Df


A Polynomial function in variable x of degree n

is of the form

k)x(f

01

2

2

1n

1n

n

n axaxaxaxa)x(P

and0a,Rs'awhere ni

.egerintnegativenonaisn

Constant function - polynomial of degree 0.

bax)x(f Linear function - polynomial of degree 1.

cbxax)x(f2 Quadratic function - polynomial of degree 2.

xx2x2)x(f2334 polynomial of degree 4

5xx3x2)x(g36 x5xx3x2)x(h

235

g and h are NOT POLYNOMIAL FUNCTIONS.Prepared by: Ms Sonia Tan

PIECE-WISE DEFINED FUNCTIONS

For y = -2x + 2

R:Df

0y:Rf

1xif1x

1xif2x2)x(f

2

When x = 1 , y = 0 (1, 0)

When x = 0 , y = 2 (0, 2)

For y = x2 + 1

When x = 1 , y = 2 (1, 2)

When x = 2 , y = 5 (2, 5)

•

y = -2x + 2

y = x2 + 1

•

•


y = 2 for x < -1

R:Dg

y = - 3 for -1 x < 2

For y = x

When x = 2 , y = 2 (2, 2)

When x = 4 , y = 4 (4, 4)

•y = 2

y = -3•

•

2xifx

2x1if3

1xif2

)x(g

y = x

2yor3y:Rg

),2[3 Prepared by: Ms Sonia Tan

functions - fcampena...prepared by: ms sonia tan domain = the set of all feasible values of x for...

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