Classwork 91H
Restricting Domain on Rational Functions Using Interval & Set-Builder Notation
CRS
Objective
Algebra 2 Content
The domain of a function is the set of all possible input values (usually x), which allows the function to work.
What values can be substituted into x to get a valid output? What values will make the expression undefined?
We must exclude numbers from a rational expression’s domain that make the denominator zero.
Example 1 – Simplify each rational expression. Find all the numbers that must be excluded from the domain of
each rational expression.
3
x
a.
b. 2
x 1
x 1


TEACHER NOTES

c.
x 2  6x  5
x 2  25
d.
x 2 1
x 2  2x 1

From the previous examples, we know how to find restrictions on the domain of a rational function. Now we
will learn how to use interval and set builder notation to represent the domain of a rational expression. A set
is a collection of things, and in our case, it will represent the real numbers in the domain of a rational function
that make it true.
Here is an example of how to use set-builder notation to represent the expression x > 0:



Example 2: Write the expression in set-builder notation.
a. x  2
b. w > 3
Read (write out):
Read (write out):
c. m  1 or m > 5
d. 3  y  4
Read (write out):

Read (write out):
Example 3: Use set-builder notation to find the domain of each rational function.
9
7
a. f (x)  2
b. g(x)  2
x  6x  9
5x 15x
Set-builder notation:

Read (write out):
c. h(x) 
Read (write out):
x  7 x 2 1

x 1 3x  21

Set-builder notation:
d. f (x) 
x  3 x2  x  6

x 2  4 x 2  6x  9

Set-builder notation:
Set-builder notation:
Read (write out):
Read (write out):
What does infinity mean?
Interval Notation – Sets of real numbers can be represented using interval notation.
Example:
1 x  4
2  x  3
x 5
x 2
x  1 or
x4
x  3 or
x 1
Number Line (how
we learned in
Algebra):
Number Line Using
Interval Notation:
Interval Notation
with Symbols:
Meaning:
Example 4: Express each interval in set-builder notation and graph the interval on a number line.
a. (-1, 4]
c. 4,
b. [2.5, 4]

Directions: Express each interval in set-builder notation and graph the interval on a number line.
1) [-2, 5)
2) [1, 3.5]
3) (,1)

Example 5: Simplify each rational expression. Use interval and set-builder notation to represent the domain of
each rational function.
x
x 2
x2  9
b. g(x)  2
c. f (x)  2
a. f (x) 
x 4
x 4
x 3



Set-builder notation:
Set-builder notation:
Set-builder notation:
Graph for Interval Notation:
Graph for Interval Notation:
Graph for Interval Notation:
Interval Notation:
Interval Notation:
Interval Notation:
Directions: Simplify each rational expression. Use interval and set-builder notation to represent the domain of
each rational function.
x
x 5
x 2  25
2) g(x)  2
3) f (x)  2
1) f (x) 
x  25
x  25
x 5



Set-builder notation:
Set-builder notation:
Set-builder notation:
Graph for Interval Notation:
Graph for Interval Notation:
Graph for Interval Notation:
Interval Notation:
Interval Notation:
Interval Notation:
4)
x 2  2x  8 x  4

x2  9
x3

5)
x 2  2x 1 x 2  x  2

x3  x
3x 2  3
6)
x 2
4

2x  3 x  3


Set-builder notation:
Set-builder notation:
Set-builder notation:
Interval Notation:
Interval Notation:
Interval Notation:
7)
3
5

x 1 x 1
8)

x3
2
 2
x  x  2 x 1
2

Set-Builder:
Interval:
Set-Builder:
Interval:
Extra Resources: