Precalculus Examples

Find the Domain f(x) = square root of x^4-16x^2
Step 1
Set the radicand in greater than or equal to to find where the expression is defined.
Step 2
Solve for .
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Step 2.1
Convert the inequality to an equation.
Step 2.2
Factor the left side of the equation.
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Step 2.2.1
Factor out of .
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Step 2.2.1.1
Factor out of .
Step 2.2.1.2
Factor out of .
Step 2.2.1.3
Factor out of .
Step 2.2.2
Rewrite as .
Step 2.2.3
Factor.
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Step 2.2.3.1
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 2.2.3.2
Remove unnecessary parentheses.
Step 2.3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 2.4
Set equal to and solve for .
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Step 2.4.1
Set equal to .
Step 2.4.2
Solve for .
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Step 2.4.2.1
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 2.4.2.2
Simplify .
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Step 2.4.2.2.1
Rewrite as .
Step 2.4.2.2.2
Pull terms out from under the radical, assuming positive real numbers.
Step 2.4.2.2.3
Plus or minus is .
Step 2.5
Set equal to and solve for .
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Step 2.5.1
Set equal to .
Step 2.5.2
Subtract from both sides of the equation.
Step 2.6
Set equal to and solve for .
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Step 2.6.1
Set equal to .
Step 2.6.2
Add to both sides of the equation.
Step 2.7
The final solution is all the values that make true.
Step 2.8
Use each root to create test intervals.
Step 2.9
Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.
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Step 2.9.1
Test a value on the interval to see if it makes the inequality true.
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Step 2.9.1.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 2.9.1.2
Replace with in the original inequality.
Step 2.9.1.3
The left side is greater than the right side , which means that the given statement is always true.
True
True
Step 2.9.2
Test a value on the interval to see if it makes the inequality true.
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Step 2.9.2.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 2.9.2.2
Replace with in the original inequality.
Step 2.9.2.3
The left side is less than the right side , which means that the given statement is false.
False
False
Step 2.9.3
Test a value on the interval to see if it makes the inequality true.
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Step 2.9.3.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 2.9.3.2
Replace with in the original inequality.
Step 2.9.3.3
The left side is less than the right side , which means that the given statement is false.
False
False
Step 2.9.4
Test a value on the interval to see if it makes the inequality true.
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Step 2.9.4.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 2.9.4.2
Replace with in the original inequality.
Step 2.9.4.3
The left side is greater than the right side , which means that the given statement is always true.
True
True
Step 2.9.5
Compare the intervals to determine which ones satisfy the original inequality.
True
False
False
True
True
False
False
True
Step 2.10
The solution consists of all of the true intervals.
or or
Step 2.11
Combine the intervals.
Step 3
The domain is all values of that make the expression defined.
Interval Notation:
Set-Builder Notation:
Step 4