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, we can choose any value we want for x. Suppose we choose 11. We can put 11 into the function by substituting for 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 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 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 us that our set contains x-values.

The colon says, such that

The symbol that looks like an e (or a c sticking its tongue out) says, belongs to...

And the cursive, or script, R is short for the set of real numbers.

R, the set of real numbers.” So we read it, “The set of xsuch that x belongs to

When we put 11 in for x,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 putting some number in for x?x?

No! We say that the range of the function is the set of real numbers.

“The set of y, such that y belongs to R, the set of real numbers.” Read this:

the domain and range can be any real number. It is not always true that Sometimes mathematicians want 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,

limits the domain or range. Sometimes the function itself In this function, can x be any real number?

were 3? What would happen if x Then we would have to divide by 0. We can never divide by 0.

3 from the domain. So we would have to eliminate The domain would be,

which could not belong to the range? Can you think of a number 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 The range of the function is,

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 the square 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, what is the domain?

What is y when x is 16? The square root of 16 is 4, so y is 4 when x is 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? Did you say –4?

not –16. There is no real number we can square to get a negative number. So no negative number can belong to the domain of

so the domain of is The smallest number for which we can find a square root is 0,

Find the domain of each function:

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