When finding the domain of even-degree roots, the expression under the radical must be greater than or equal to 0.
Example 1 Find the domain of
y=x
y
x
{x|x≥0}
x
x
0
PRACTICE - Find the Domain of the following:
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Solution 1
{x|x≥52}
x
x
52
since
2x-5≥0
2x5
0
,
2x≥5
2x
5
, x≥52
x
52
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Solution 2
{x|x≤7}
x
x
7
since
7-x≥0
7x
0
,
-x≥-7
x
7
,
x≤7
x
7
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The rest of the answers will be expressed in interval notation since that is a simpler way to express answers.
Problem 3
y=4x2-164
y
4
4x216
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Solution 3
-∞-2⋃2∞
2
2
since
4x2-16≥0
4x216
0
,
4x2≥16
4
x2
16
,
x2≥4
x2
4
,
x≤-2∨x≥2
x
2
x
2
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Problem 4
y=16-25x2
y
16
25
x
2
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Solution 4
-4545
45
45
since
16-25x2≥0
16
25
x2
0
,
-25x2≥-16
-25
x2
-16
,
x2≤1625
x2
1625
,
x≥-45∧x≤45
x
4
5
x
4
5
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Problem 5
y=x-7x+1
y
x
7
x
1
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Solution 5
-∞-1⋃7∞
1
7
,
x-7x+1≥0
x
7
x
1
0
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Problem 6
y=2x2-7x+3
y
2
x
2
7
x
3
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Solution 6
-∞1/2⋃3∞
12
3
,
2x2-7x+3≥0
2
x
2
7
x
3
0
,
2x-1x-3≥0
2
x
1
x
3
0
,
x≤12∨x≥3
x
1
2
x
3
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Problem 7
y=xx2+4
y
x
x
2
4
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Solution 7
-∞∞
,
since
x2+4≥0
x
2
4
0
,
x2≥-4
x
2
4
This will always be true, for all real numbers, any number squared is always positive
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Problem 8
y=x+-x+8
y
x
x8
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Solution 8
-∞8
8
since
-x+8≥0
x
8
0
,
-x≥-8
x
8
,
x≤8
x
8
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Solution 9
-∞∞
,
since
6x2+8≥0
6
x
2
8
0
,
6x2≥-8
6
x
2
8
,
x2≥-86
x
2
-86
This will always be true, for all real numbers, any number squared is always positive
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Problem 10
y=-8-6x2
y
86x2
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Solution 10
No solution since
-8-6x2≥0
8
6
x2
0
,
-6x2≥8
-6
x2
8
,
x2≥-86
x2
-86
This will never be true, so there is no solution, since any number squared is always positive, so it will never be less than 0.
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