math 8

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. Given the function,
,
5
2 
 x
y
we can choose any value
we want for x.
Suppose we choose 11.
We can put 11 into the function
by substituting for x.

5. 5
22  5
2 
x 5
)
11
(
2 

y 17
11
If 
x

6. 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. In our function
,
5
2 
 x
y
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
 


x
x :
Does that look like a foreign
language?
Let’s translate:

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

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

13. The colon says,
such that
 
:
x

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

15. And the cursive, or script,
R
 


x
x:
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
x 

x

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

18. So 17 belongs to
the range of the function,
Is there any number that
we could not get for y by
.
5
2 
 x
y
putting some number in for x?

19. No!
We say that the range of
the function is
5
2 
 x
y
the set of real numbers.

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

21. 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.

22. the function
They might think about
where x is between –3 and 3.
It could be written,
4
2

 x
y
 
3
3
4
2




 x
x
y

23. limits the domain or range.
Sometimes the function itself
In this function,
3
1


x
y
can x be any real number?

24. were 3?
What would happen if x
Then we would have to
divide by 0.
3
1


x
y
We can never
divide by 0.

25. 3 from the domain.
So we would have to eliminate
3
1


x
y
The domain would be,
 
3
: 
x
x

26. which could not belong to the
range?
Can you think of a number
3
1


x
y
y could never be 0.
Why?

27. 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
3
1


x
y
The range of the function is,
 
0
: 
y
y

28. 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.

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

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

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

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

33. not –16.
     ,
16
4
4
4 2





There is no real number we can
square to get a negative number.
So no negative number can belong
to the domain of
x
y 

34. so the domain of
is
x
y 
The smallest number for which
we can find a square root is 0,
 
0
: 
x
x

35. Find the domain of each function:
5
1
.
1


x
y
17
4
.
2 
 x
y
9
.
3 
 x
y
99
.
4 2

 x
y

36. Answers:
 
5
:
.
1 

x
x
 


x
x :
.
2
 


x
x :
.
4
 
9
:
.
3 

x
x