FUNCTIONS

Relation – a set of ( x , y ) points

Function – a set of ( x , y ) points where there is only

one output for each specific input

– x can not be paired with more than one y

** just make sure x doesn’t repeat itself

Ways to represent functions :

1. A set of ( x , y ) coordinates

2. A mapping

3. An equation

FUNCTIONS

Relation – a set of ( x , y ) points

Function – a set of ( x , y ) points where there is only

one output for each specific input

– x can not be paired with more than one y

** just make sure x doesn’t repeat itself

Ways to represent functions :

1. A set of ( x , y ) coordinates

2. A mapping

3. An equation

The DOMAIN of a function are its x values, its RANGE are the y values

This relation represents a true function, notice that the x coordinate never repeats…

( 1 , 3 ) , ( 2 , 4 ) , ( - 5 , - 3 ) , ( 0 , 2 )

This relation represents a true function, notice that the x coordinate never repeats…

( 1 , 3 ) , ( 2 , 4 ) , ( - 5 , - 3 ) , ( 0 , 2 )

Lets look at a mapping. Mapping has the x’s on one side and the y’s on the other side. You draw arrows showing which x is associated with which y.

This relation represents a true function, notice that the x coordinate never repeats…

( 1 , 3 ) , ( 2 , 4 ) , ( - 5 , - 3 ) , ( 0 , 2 )

Lets look at a mapping. Mapping has the x’s on one side and the y’s on the other side. You draw arrows showing which x is associated with which y.

Let’s use the ( x , y ) pairs from the above example:

x y

1

2

-5

0

2

- 3

4

3

This relation represents a true function, notice that the x coordinate never repeats…

( 1 , 3 ) , ( 2 , 4 ) , ( - 5 , - 3 ) , ( 0 , 2 )

Lets look at a mapping. Mapping has the x’s on one side and the y’s on the other side. You draw arrows showing which x is associated with which y.

Let’s use the ( x , y ) pairs from the above example:

x y

1

2

-5

0

2

- 3

4

3

This relation represents a true function, notice that the x coordinate never repeats…

( 1 , 3 ) , ( 2 , 4 ) , ( - 5 , - 3 ) , ( 0 , 2 )

Lets look at a mapping. Mapping has the x’s on one side and the y’s on the other side. You draw arrows showing which x is associated with which y.

Let’s use the ( x , y ) pairs from the above example:

x y

1

2

-5

0

2

- 3

4

3

This relation represents a true function, notice that the x coordinate never repeats…

( 1 , 3 ) , ( 2 , 4 ) , ( - 5 , - 3 ) , ( 0 , 2 )

Lets look at a mapping. Mapping has the x’s on one side and the y’s on the other side. You draw arrows showing which x is associated with each y.

Let’s use the ( x , y ) pairs from the above example:

x y

1

2

-5

0

2

- 3

4

3

x y

1

2

-1

-2

1

4

Does the mapping show a true function ?

x y

1

2

-1

-2

1

4

Does the mapping show a true function ?

YES !!! Each x has only one y in the mapping

** its acceptable for y to have more than one match

x y

2

-2

1

-3

4

Does the mapping show a true function ?

x y

2

-2

1

-3

4

Does the mapping show a true function ?

NO !!! Notice that 2 has two matches.

x y

2

-2

1

-3

4

Does the mapping show a true function ?

NO !!! Notice that 2 has two matches.

If we showed the mapping as coordinates, you see that x repeats.

( 2 , 1 ) , ( 2 , 4 ) , ( - 2 , - 3 )

FUNCTIONS :

There is a special notation to show a function.

)(xf

FUNCTIONS :

There is a special notation to show a function.

)(xf - Read “f of x “

- There is a function f that has x as its variable

- it’s a different way of saying ”y”

- coordinate is ( x , f(x) )

FUNCTIONS :

There is a special notation to show a function.

53)( xxf

The rule of the function

FUNCTIONS :

There is a special notation to show a function.

53)( xxf

We can generate some ( x , f(x) ) points by picking some values for x, plugging them into the rule and solving for f(x)

FUNCTIONS :

There is a special notation to show a function.

53)( xxf

We can generate some ( x , f(x) ) points by picking some values for x, plugging them into the rule and solving for f(x)

When substituting a value for x, you show it in the f(x) notation…lets use x = 1

x f(x)

FUNCTIONS :

There is a special notation to show a function.

53)( xxf

We can generate some ( x , f(x) ) points by picking some values for x, plugging them into the rule and solving for f(x)

When substituting a value for x, you show it in the f(x) notation…lets use x = 1

x f(x)

1

8)1(

5)1(3)1(

53)(

f

f

xxf

8

FUNCTIONS :

There is a special notation to show a function.

53)( xxf

We can generate some ( x , f(x) ) points by picking some values for x, plugging them into the rule and solving for f(x)

When substituting a value for x, you show it in the f(x) notation…lets use x = 2

x f(x)

1

11)2(

5)2(3)2(

53)(

f

f

xxf

8

2 11

FUNCTIONS :

There is a special notation to show a function.

53)( xxf

We can generate some ( x , f(x) ) points by picking some values for x, plugging them into the rule and solving for f(x)

When substituting a value for x, you show it in the f(x) notation…lets use x = 3

x f(x)

1

14)3(

5)3(3)3(

53)(

f

f

xxf

8

2 11

3 14

FUNCTIONS :

There is a special notation to show a function.

53)( xxf

We can generate some ( x , f(x) ) points by picking some values for x, plugging them into the rule and solving for f(x)

You could keep going here, 3 points is enough for a linear function.

x f(x)

1 8

2 11

3 14

FUNCTIONS :

There is a special notation to show a function.

53)( xxf

We can generate some ( x , f(x) ) points by picking some values for x, plugging them into the rule and solving for f(x)

Does this set of points satisfy a true function ?

x f(x)

1 8

2 11

3 14

FUNCTIONS :

There is a special notation to show a function.

53)( xxf

We can generate some ( x , f(x) ) points by picking some values for x, plugging them into the rule and solving for f(x)

Does this set of points satisfy a true function ?

YES…

x f(x)

1 8

2 11

3 14

FUNCTIONS :

62)2()2(

12)1()1(

22)0()0(

32)1()1(

102)2()2(

2)(

3

3

3

3

3

3

f

f

f

f

f

xxf

When generating points for quadratics, cubics, etc, it is a good idea to get 5 – 8 points ( + / - ) to show your relation and to eventually graph your function…

Below is the work to generate the ( x , f(x) ) coordinates for the given function…

x f(x)

-2

-1

0

1

2

-10

-3

-2

-1

6

FUNCTIONS :

When evaluating functions, sometimes algebraic expressions are used…

4)( 2 xxf

Complete the table below…

x )(xf

FUNCTIONS :

When evaluating functions, sometimes algebraic expressions are used…

4)( 2 xxf

Complete the table below…

4

x )(xf

FUNCTIONS :

When evaluating functions, sometimes algebraic expressions are used…

4)( 2 xxf

Complete the table below…

4 12

x )(xf 124

4164

444

42

2

f

f

f

xxf

FUNCTIONS :

When evaluating functions, sometimes algebraic expressions are used…

4)( 2 xxf

Complete the table below…

4 12

a

x )(xf

FUNCTIONS :

When evaluating functions, sometimes algebraic expressions are used…

4)( 2 xxf

Complete the table below…

4 12

a a2 – 4

x )(xf

4

4

4

2

2

2

aaf

aaf

xxf

FUNCTIONS :

When evaluating functions, sometimes algebraic expressions are used…

4)( 2 xxf

Complete the table below…

4 12

a a2 – 4

x )(xf

3m

FUNCTIONS :

When evaluating functions, sometimes algebraic expressions are used…

4)( 2 xxf

Complete the table below…

4 12

a a2 – 4

x )(xf

3m

563

4963

433

4

2

2

2

2

mmmf

mmmf

mmf

xxf

562 mm

FUNCTIONS :

When evaluating functions, sometimes algebraic expressions are used…

4)( 2 xxf

Complete the table below…

4 12

a a2 – 4

2x + h

x )(xf

3m

hxh

hxh

h

hxh

h

xhxhx

h

xhxhx

h

xhx

h

xfhxf

xxf

2

22

442

442

44

4

2

222

222

22

2

562 mm

h

xfhxf