1. Domain
and
Range

2. As we study functions
we learn terms like
input values
and
output values.

3. 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.

4. we can choose any value we want for x.
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.

6. we would have the set of numbers we call the domain of the function.
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.

7. The domain is the set that contains all the input values for a function.

8. is there any number we could not put in for x?
In our function
is there any number we could not put in for x?
No!

9. Because we could substitute any real number for x,
we say the domain of the function is the set of real numbers.

10. To use the symbols of algebra, we could write the domain as
Let’s translate:
Does that look like a foreign language?

11. just tell us we have a set of numbers.
The curly braces
just tell us we have a set of numbers.

12. that our set contains x-values.
The x reminds us
that our set contains x-values.

13. The colon says,
such that

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

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

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

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

18. the range of the function,
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?

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

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

21. It is not always true that the domain and range
can be any real number.
Sometimes mathematicians
want to study a function over
a limited domain.

22. They might think about
the function
where x is between –3 and 3.
It could be written,

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

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

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

26. Can you think of a number which could not belong to the range?
Why?
y could never be 0.

27. The range of the function is,
What would x have to be
for y to be 0?
The range of the function is,
There is no number we can divide 1 by to get 0, so 0 cannot belong to the range.

28. The most common rules of algebra
that limit the domain of functions are:
Rule 1: You can’t divide by 0.
Rule 2: You can’t take the
square root of a negative number.

29. We’ve already seen an example of Rule 1: You can’t divide by 0.

30. You can’t take the square root of a negative number.
Think about Rule 2,
You can’t take the square root of a negative number.
Given the function,
what is the domain?

31. and 4 belongs to the range.
What is y when x is 16?
The square root of 16 is 4,
so y is 4 when x is 16
16 belongs to the domain,
and 4 belongs to the range.

32. What number do you square to get –16?
But what is y when x is –16?
What number do you square to get –16?
Did you say –4?

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

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

35. Find the domain of each function:

36. Answers: