Finding the Domain of Radical Functions
Summary: Finding the domain of radical/root functions.
When finding the domain of even-degree roots, the expression under the radical must be greater than or equal to 0.
Example 1
{ x | x ≥ 0 }
x
x
0
Find the domain of
y = x
y
x
PRACTICE - Find the Domain of the following:
Problem 1
y = 2 x - 5
y
2 x 5
Solution 1
{ x | x ≥ 5 2 }
x
x
5 2
2 x - 5 ≥ 0
2 x 5 0
2 x ≥ 5
2 x 5
x ≥ 5 2
x
5 2
Problem 2
y = 7 - x 4
y
4 7 x
Solution 2
{ x | x ≤ 7 }
x
x
7
7 - x ≥ 0
7 x 0
- x ≥ - 7
x 7
x ≤ 7
x
7
The rest of the answers will be expressed in interval notation since that is a simpler way to express answers.
Problem 3
y = 4 x 2 - 16 4
y
4 4 x 2 16
Solution 3
- ∞ - 2 ⋃ 2 ∞
2 2
4 x 2 - 16 ≥ 0
4 x 2 16 0
4 x 2 ≥ 16
4
x 2 16
x 2 ≥ 4
x 2 4
x ≤ - 2 ∨ x ≥ 2
x
2
x
2
Problem 4
y = 16 - 25 x 2 y
16
25
x
2
Solution 4
- 4 5 4 5
4 5 4 5
16 - 25 x 2 ≥ 0
16
25
x 2 0
-25 x 2 ≥ -16
-25
x 2 -16
x 2 ≤ 16 25
x 2 16 25
x ≥ - 4 5 ∧ x ≤ 4 5 x
4 5
x
4
5
Problem 5
y = x - 7 x + 1
y
x
7
x
1
Solution 5
- ∞ - 1 ⋃ 7 ∞
1 7
x - 7 x + 1 ≥ 0
x
7
x
1
0
Problem 6
y = 2 x 2 - 7 x + 3
y
2
x
2
7
x
3
Solution 6
- ∞ 1 / 2 ⋃ 3 ∞
1 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
x ≤ 1 2 ∨ x ≥ 3 x
1
2
x
3
Problem 7
y = x x 2 + 4 y
x
x
2
4
Solution 7
- ∞ ∞
x 2 + 4 ≥ 0 x
2
4
0
x 2 ≥ - 4 x
2
4
Problem 8
y = x + - x + 8
y
x
x 8
Solution 8
- ∞ 8 8
- x + 8 ≥ 0 x 8
0
- x ≥ - 8 x 8
x ≤ 8 x
8
Problem 9
y = 6 x 2 + 8
y
6 x 2 8
Solution 9
- ∞ ∞
6 x 2 + 8 ≥ 0 6
x
2
8
0
6 x 2 ≥ - 8 6
x
2
8
x 2 ≥ -8 6 x
2
-8 6
Problem 10
y = - 8 - 6 x 2
y
8 6 x 2
Solution 10
No solution since
- 8 - 6 x 2 ≥ 0 8 6
x 2 0
-6 x 2 ≥ 8 -6
x 2 8
x 2 ≥ -8 6 x 2 -8 6
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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 Apr 27, 2006 3:34 pm GMT-5.