Finding the Domain of Simple Rational Functions
Summary: Finding the domain of rational functions with no other complications such as roots/radicals.
For fractions, the denominator (the bottom) of the fraction cannot equal 0 0 0 0
Example 1 { x | x ≠ 0 } x x 0
Find the domain of y = 1 x y 1 x
Problem 1
Find the domain of y = 1 x - 5 y 1 x 5
Solution 1{ x | x ≠ 5 } x x 5 x - 5 ≠ 0 x 5 0 x ≠ 5 x 5
Problem 2
Find the domain of
y = 4 x + 3 x - 7
y
4
x
3
x
7
Solution 2
{ x | x ≠ 7 } x x 7 x - 7 ≠ 0 x 7 0 x ≠ 7 x 7
Problem 3
Find the domain of
y = 7 x 5 - 2 x y 7 x 5 2 x
Solution 3
{ x | x ≠ 5 2 } x x 5 2 5 - 2 x ≠ 0 5 2 x 0 x ≠ 5 2 x 5 2
Problem 4
Find the domain of
y = 2 x - 3 x + 7
y
2
x
3
x
7
Solution 4
{ x | x ≠ 3 ∨ - 7 }
x
x
3
7
x ≠ 3
x
3
x ≠ - 7
x
7
Problem 5
Find the domain of
y = 7 x 2 x 2 - 7 x + 3
y
7
x
2
x
2
7
x
3
Solution 5
{ x | x ≠ 1 2 ∨ 3 }
x
x
1 2
3
2 x 2 - 7 x + 3 ≠ 0
2
x
2
7
x
3
0
2 x - 1 x - 3 ≠ 0
2 x 1
x
3
0
2 x - 1 ≠ 0
2 x 1
0
x - 3 ≠ 0
x
3
0
x ≠ 1 2
x
1 2
x ≠ 3
x
3
Problem 6
y = 2 x + 1 x + 5 2
y
2
x
1
x
5
2
-1
Solution 6
{ x | x ≠ - 5 }
x
x
5
x + 5 2 ≠ 0
x 5 2 0
x + 5 ≠ 0
x 5 0
x ≠ - 5
x
5
Problem 7
Find the domain of
y = x + 3 x 2 + 25
y
x
3
x
2
25
-1
Solution 7
{ x | x ∈ ℝ }
x
x
x 2 + 25 ≠ 0
x
2
25
0
x 2 ≠ - 25
x 2 25
x ∈ ℝ
x
Problem 8
Find the domain of
y = x - 7 x 2 + 2
y
x
7
x
2
2
-1
Solution 8
{ x | x ∈ ℝ }
x
x
x 2 + 2 ≠ 0
x
2
2
0
x 2 ≠ - 2
x 2 2
x ∈ ℝ
x
Problem 9
Find the domain of
y = 5 | x - 3 |
y
5
x 3
Solution 9
{ x | x ≠ 3 } x x 3
| x - 3 | ≠ 0
x 3 0
x - 3 ≠ 0
x 3
0
x ≠ 3
x
3
Problem 10
Find the domain of
y = 4 | x | - 4
y
4
x 4
Solution 10
{ x | x ≠ - 4 ∨ 4 }
x
x
4 4
| x | - 4 ≠ 0
x
4
0
| x | ≠ 4
x
4
x ≠ -4
x
-4
x ≠ 4
x
4
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This work is licensed by Pradnya Bhawalkar and Kim Johnston. See the Creative Commons License about permission to reuse this material.
Last edited by Pradnya Bhawalkar on May 1, 2006 6:47 pm GMT-5.