As we study functions

we learn terms like

input values

and

output values.

Input values are the numbers

we put into the function.

They are the x-values.

Output values are the numbers

that come out of the function.

They are the y-values.

Given the function, ,52 xy

we can choose any value we want for x.

Suppose we choose 11.

We can put 11 into the function by substituting for x.

522 52 x 5)11(2 y 17

11 If x

If we wrote down every number we could put in for x and still have the function

make sense,we would have the set of numbers we call the domain

of the function.

The domain is the set that contains all the

input values for a function.

In our function ,52 xy

is there any number we could not put in for x?

No!

Because we could substitute any real number

for x,

we say the domain of the function is the set of real numbers.

To use the symbols of algebra, we could write the domain as

xx :Does that look like a foreign

language?Let’s translate:

The curly braces just tell us we have a set of

numbers.

The x reminds usthat our set contains x-values.

x

The colon says,such that

:x

: xx

The symbol that looks like an e(or a c sticking its tongue out)

says, belongs to . . .

And the cursive, or script, R

xx:

is short for the set of real numbers.

R, the set of real numbers.”

So we read it, “The setof x

:

such that x belongs to

x x

17y

11 If x

When we put 11 in for x,y was 17.

So 17 belongs to the range of the function,

Is there any number that

we could not get for y by

.52 xy

putting some number in for x?

No!

We say that the range of

the function is

52 xy

the set of real numbers.

“The set of y, such that

y belongs to R,

the set of real numbers.”

Read this:

yy :

the domain and range can be any real number.

It is not always true that

Sometimes mathematicianswant to study a function over

a limited domain.

the function

They might think about

where x is between –3 and 3.

It could be written,

42 xy

3342 xxy

limits the domain or range.

Sometimes the function itself

In this function,

31

x

y

can x be any real number?

were 3?

What would happen if x

Then we would have to divide by 0.

31

x

y

We can never divide by 0.

3 from the domain.

So we would have to eliminate

31

x

y

The domain would be,

3: xx

which could not belong to the range?

Can you think of a number

31

x

y

y could never be 0.Why?

There is no number we can divide 1 by to get 0, so 0 cannot

belong to the range.

for y to be 0?

What would x have to be

31

x

y

The range of the function is,

0: yy

that limit the domain of functions are:

The most common rules of algebra

Rule 1: You can’t divide by 0.

Rule 2: You can’t take thesquare root of a negative number.

of Rule 1: You can’t divide by 0.

We’ve already seen an example

You can’t take the square root of a negative number.

Think about Rule 2,

Given the function,,xy

what is the domain?

What is y when x is 16?

The square root of 16 is 4,

xy

so y is 4 when x is 16

16y

16 belongs to the domain,and 4 belongs to the range.

But what is y when x is –16?

What number do you square to get –16?

xy 16y

Did you say –4?

not –16. ,16444 2

There is no real number we can square to get a negative number.

So no negative number can belong to the domain of

xy

so the domain of

isxy

The smallest number for which we can find a square root is 0,

0: xx

Find the domain of each function:

51

.1

x

y

174.2 xy

9.3 xy

99.4 2 xy

Answers:

5:.1 xx

xx :.2

xx :.4

9:.3 xx