Download R1. Rational Expressions: the Domain

R1. Rational Expressions: the Domain
Lecture 1
Consider the following fractions. Try each one, then check with the solutions
below.
1. Simplify 42 .
2. Simplify
33
51 .
3. Simplify 40 .
4. Simplify 13 .
5. Simplify 03 .
6. Simplify
42
2−4( 21 )
7. Simplify
2x+3x
.
5
8. Simplify
4x
3−3 .
Check your answers.
= 21 .
1.
2
4
2.
33
51
3.
0
4
= 0.
4.
3
1
= 3.
5.
3
0
is undefined.
6.
42
2−4( 21 )
7.
2x+3x
5
8.
4x
3−3
=
3·11
3·17
=
=
=
4x
0
=
11
17 .
16
2−2
5x
5
=
16
0
which is undefined.
= x.
which is undefined.
When is the fraction undefined? In the above examples, the fraction was undefined if the denominator either was zero, or simplified to zero. Consider three
more fractions. In each problem, ask, “What would make the fraction undefined?”
The fraction
3x + 1
x−5
1
would be undefined when the denominator is zero. This leads to the question,
“When is x − 5 = 0?” The answer to the question comes from solving x − 5 = 0.
The solution is x = 5.
The fraction
3
2−x
would be undefined when the denominator is zero. This leads to the question,
“When is 2 − x = 0?” The answer to the question comes from solving 2 − x = 0.
The solution is x = 2.
The fraction
3x + 2
(x + 2)(x − 3)
would also be undefined when the denominator is zero. This leads to the question, “When is (x + 2)(x − 3) = 0?” The answer to the question comes from
solving (x + 2)(x − 3) = 0. This problem might look more difficult, but it is exactly the kind of problem that was solved at the end of the chapter on factoring.
This problem is solved by setting each factor to zero and solving.
x−3 = 0
+3 = +3
x = 3
x+2 = 0
−2 = −2
x = −2
Verify that the answers above make the fractions undefined, i.e., the denominator is zero. Following is the check for each of the above three examples.
When x = 5.
3x + 1
3(5) + 1
=
x−5
5−5
16
=
0
Undefined, as expected.
When x = 2.
3
3
=
2−x
2−2
3
=
0
Undefined, as expected.
2
First when x = −2, then when x = 3.
3(−2) + 2
3(x) + 2
=
(x + 2)(x − 3)
(−2 + 2)(−2 − 3)
−6 + 2
=
(0)(−5)
−4
=
0
Undefined, as expected.
When x = 3.
3(x) + 2
3(3) + 2
=
(x + 2)(x − 3)
(3 + 2)(3 − 3)
9+2
=
(5)(0)
11
=
0
Definition. The domain of an expression is any number that does not make
the expression undefined. With this definition in mind, what is the domain of
the above three examples? Before reading, see if you can answer this.
For the fraction
3x+1
x−5
For the fraction
3
2−x
For the fraction
3x+2
(x+2)(x−3)
the domain is any number except x = 5.
the domain is any number except x = 2.
the domain is any number except x = −2 or x = 3.
These sentences can be written using set builder notation.
“The domain is any number except x = 5” can be written in set builder notation
as
{x|x 6= 5}
“The domain is any number except x = 2” can be written in set builder notation
as
{x|x 6= 2}
“The domain is any number except x = −2 or x = 3” can be written in set
builder notation as
{x|x 6= −2, 3}
3
You can either write the sentence or use the set builder notation. They express
the same message; however, as you progress in mathematics, you’ll tend to see
the set builder notation more often.
x2 − 1
.
x2 − 7x
Example. Find the domain of
Solution. As before, the only concern is when the denominator is zero. This
leads to the question, “When is x2 − 7x = 0?” The answer lies in solving that
equation.
x2 − 7x = 0
x(x − 7) = 0
x−7 = 0
+7 = +7
x = 7
x =0
x =0
x2 − 1
is any number except x = 0 or x = 7. Or, in set
x2 − 7x
builder notation, {x|x 6= 0, 7}.
Thus, the domain of
0.1
Practice Problems
Find the set of Real numbers for which each of the following rational expressions
is defined, i.e., find the domain of the rational expressions. Write your answer
in set builder notation.
1.
3+x
x−4
2.
x2 − 6
x
3.
4x − 7
x2 + 5
4.
3x − 1
x2 − 7x
5.
x2 − 1
x2 + 6x
6.
x2 + 3x − 10
x2 + 7x + 10
7.
x−5
x2 − 3x − 28
8.
3x − 8
4x + 20
9.
5
x2 − 9
10.
x−3
x2 + 4
11.
3x2 − 6x + 4
2x2 + 3x − 2
Additional Thoughts
4
12.
x2 + 7x − 2
6
Two types of problems cause some consternation. These examples were not
initially provided with the notes in order to allow the student the opportunity
to trust what was learned and be challenged. Remember to ask the question
regarding the denominator, trust the math, and think about what the math
communicates.
Example. Find the domain of
2x − 5
6
Remember, the question stays the same, “When does 6 = 0?” This may seem
a strange question, because 6 does not equal 0. The math is telling us that
the denominator can never be zero. Thus, the domain is all real numbers, or
{x|x is any real number}.
Example. Find the domain of
3x − 5
x2 + 8
Again, set the denominator to zero. This gives x2 + 8 = 0. Solve this by first
playing the game
8
0
After some exploration, one finds that there is no solution to this game. Since
there is no solution to the game, the equation has no solution, i.e., there is no
way to make the denominator equal to zero. Illustrated another way,
x2 + 8 = 0
x2 = −8.
The last line is asking, “What number squared gives −8”. We know that all real
numbers, when squared, are always positive. Logic dictates that since no number
squared is negative, there cannot be a solution. Since no number can make the
denominator zero, the domain is all real numbers, or {x|x is any real number}.
5
0.2
Solutions
1. {x | x 6= 4}
2. {x | x 6= 0}
3. All real numbers.
4. {x | x 6= 0, 7}
5. {x | x 6= −6, 0}
6. {x | x 6= −5, −2}
7. {x | x 6= −4, 7}
8. {x | x 6= −5}
9. {x | x 6= −3, 3}
10. All real numbers.
11. {x | x 6= −2, 21 }
6
12. All real numbers.