Name: _________________________________________ Date: ______ Period:_________
R7 Domain and Rational Expressions Notes
Part 1: Domain of functions
In determining the domain of a function, there are two main restrictions to look out for.
Exception 1) The denominator of a fraction can't equal _________.
Exception 2) A radicand with an even index can't be ______________.
Set-Builder Notation:
This is read "all x that are elements of the set of integers, such that, x is greater than 0."






Important symbols to learn:
-  is the set of natural numbers 1, 2, 3, ...
-  denotes the integers 0, 1, -1, 2, -2, ....
-  denotes the set of rational numbers (above with fractions).
-  denotes the set of all real numbers, consisting of all rational numbers and
irrational numbers
- denotes the set of all complex numbers.
- ∅ is the empty set, the set which has no elements.
Determine the domain of the following:
1
x5
a) g(x) = 3x2 + 4x + 5
b) h( x) 
Inequality Notation: ______________
Inequality Notation: ______________
Interval Notation: ________________
Interval Notation: ________________
Set-builder Notation: _____________
c) f ( x)  3 x  1
Set-builder Notation: _____________
d) k ( x)  4  3 x
e) f ( x) 
5
8

x x 3
f) g ( x) 
x 5
x 5
Part 2: Rational Expressions

A "rational expression" is a ________________ of two polynomials.

A rational expression is in simplified form if its numerator and denominator have no common
factors other than _________.
Simplifying Rational Expressions (Fractions):
1. Factor all expressions (numerators and denominators)
2. Reduce by canceling out ___________ ____________ (terms in parentheses) from the
numerator and denominator
4 x
3. Look to cancel opposites; leave a (-1) behind. Ex:
=
x4
4. Re-write answer as a new and reduced fraction
Example 1: Simplify each rational expression.
a)
Try It!
x2  4
x 2  9 x  14
b)
2 x 2  2 x  40
x 2  2 x  15
3x3  6 x 2  12 x
x3  8
1.
3.
x 2  11x  24
x 2  3x  40
2.
x3  5 x 2  3x  15
x 2  8 x  15
3x 2  108
4. 2
x  12 x  36
Multiplying/Dividing Rational Expressions:
1. Factor all expressions (numerators and denominators) _______________
2. If a division problem, _________ the second fraction and write as a ___________________ problem.
3. Multiply the _______________ together and multiply the ________________ together (just write
factors next to each other; don’t actually FOIL)
4. Reduce by canceling out ___________ ____________ (terms in parentheses) from the numerator
and denominator
5. Re-write answer as a new and reduced fraction
Example 2
2 x  4 x x  9 x 18
2x
x  4 x 12
2
2
Practice
2
Example 3
3
8x2  8x
 2
x  7 x  6x  7
7.
48x5 y 3 x 2 y
y 4 6 x3 y 2
8.
x 2  3x  10 2
( x  10 x  21)
x 2  2 x  15
9.
4  x  5  x  x  1
x2
2  x  5
10.
 y2  4
x3  8
 y  2  x2  2x  4
11.
 x  2  x 2  9 x  14


2  x  7
12.
8 x 2 y 2 z 10 xy
 4
xz 3
x z
13.
x2  4 x  5
  x 2  6 x  5
x5
x 2  8 x  15
14.
  x 2  x  20 
x2  4x
16.
x 2  100 x3  10 x 2
x
4 x2
x
15.
17.
x
2
 x  20 
x5
  x  4
x2  5x
2

x5
x4
When adding or subtracting rational expressions with LIKE denominators, we add or
subtract the numerators only (just like in elementary school!).
Example 1:
Example 2:
7 2
+ =
6x 6x
(
5
3x - 4
=
x+2
x+2
) (
)
Try it:
a)
x
4

2
16 x 16 x 2
d)
2x
8
 3
x 3 x 3
3
b)
3 y2
5x

x5 x5
e)
c)
17
12

2x  3 2x  3
4
n

x3 x3
When adding or subtracting with UNLIKE denominators, we have to first find the LCM
(least common multiple). Figure out what each denominator is missing and then multiply.
***REMEMBER*** What you do to the bottom you must do to the top!
Example 3:
Example 4:
7
x
 2
2
9x
3x  3x
12
3

x  5 x  24 x  3
2
Try It!
a)
4
2

x 3 x 6
b)
3x
6
2


x  x  12 x  4 x  3
c)
4
2x
 2
x  5 x  25
d)
3x
4x

x  2 4  x  2
2
A ______________ _______________ contains a fraction in its numerator or denominator.
Example 5:
Example 6:
5
x4
1
2

x4 x
1
5 2

2x x
Try it:
a)
2
3

x x 1
1
2x  2
b)
3
2

x2 3
2x 1

x2 x