Trigonometry Examples

Find the Inverse x=y^2-2y
Step 1
Rewrite the equation as .
Step 2
Subtract from both sides of the equation.
Step 3
Use the quadratic formula to find the solutions.
Step 4
Substitute the values , , and into the quadratic formula and solve for .
Step 5
Simplify.
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Step 5.1
Simplify the numerator.
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Step 5.1.1
Raise to the power of .
Step 5.1.2
Multiply .
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Step 5.1.2.1
Multiply by .
Step 5.1.2.2
Multiply by .
Step 5.1.3
Factor out of .
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Step 5.1.3.1
Factor out of .
Step 5.1.3.2
Factor out of .
Step 5.1.4
Rewrite as .
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Step 5.1.4.1
Rewrite as .
Step 5.1.4.2
Rewrite as .
Step 5.1.5
Pull terms out from under the radical.
Step 5.1.6
One to any power is one.
Step 5.2
Multiply by .
Step 5.3
Simplify .
Step 6
Simplify the expression to solve for the portion of the .
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Step 6.1
Simplify the numerator.
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Step 6.1.1
Raise to the power of .
Step 6.1.2
Multiply .
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Step 6.1.2.1
Multiply by .
Step 6.1.2.2
Multiply by .
Step 6.1.3
Factor out of .
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Step 6.1.3.1
Factor out of .
Step 6.1.3.2
Factor out of .
Step 6.1.4
Rewrite as .
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Step 6.1.4.1
Rewrite as .
Step 6.1.4.2
Rewrite as .
Step 6.1.5
Pull terms out from under the radical.
Step 6.1.6
One to any power is one.
Step 6.2
Multiply by .
Step 6.3
Simplify .
Step 6.4
Change the to .
Step 7
Simplify the expression to solve for the portion of the .
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Step 7.1
Simplify the numerator.
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Step 7.1.1
Raise to the power of .
Step 7.1.2
Multiply .
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Step 7.1.2.1
Multiply by .
Step 7.1.2.2
Multiply by .
Step 7.1.3
Factor out of .
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Step 7.1.3.1
Factor out of .
Step 7.1.3.2
Factor out of .
Step 7.1.4
Rewrite as .
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Step 7.1.4.1
Rewrite as .
Step 7.1.4.2
Rewrite as .
Step 7.1.5
Pull terms out from under the radical.
Step 7.1.6
One to any power is one.
Step 7.2
Multiply by .
Step 7.3
Simplify .
Step 7.4
Change the to .
Step 8
The final answer is the combination of both solutions.
Step 9
Interchange the variables. Create an equation for each expression.
Step 10
Solve for .
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Step 10.1
Rewrite the equation as .
Step 10.2
Subtract from both sides of the equation.
Step 10.3
To remove the radical on the left side of the equation, square both sides of the equation.
Step 10.4
Simplify each side of the equation.
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Step 10.4.1
Use to rewrite as .
Step 10.4.2
Simplify the left side.
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Step 10.4.2.1
Simplify .
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Step 10.4.2.1.1
Multiply the exponents in .
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Step 10.4.2.1.1.1
Apply the power rule and multiply exponents, .
Step 10.4.2.1.1.2
Cancel the common factor of .
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Step 10.4.2.1.1.2.1
Cancel the common factor.
Step 10.4.2.1.1.2.2
Rewrite the expression.
Step 10.4.2.1.2
Simplify.
Step 10.4.3
Simplify the right side.
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Step 10.4.3.1
Simplify .
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Step 10.4.3.1.1
Rewrite as .
Step 10.4.3.1.2
Expand using the FOIL Method.
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Step 10.4.3.1.2.1
Apply the distributive property.
Step 10.4.3.1.2.2
Apply the distributive property.
Step 10.4.3.1.2.3
Apply the distributive property.
Step 10.4.3.1.3
Simplify and combine like terms.
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Step 10.4.3.1.3.1
Simplify each term.
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Step 10.4.3.1.3.1.1
Multiply by .
Step 10.4.3.1.3.1.2
Move to the left of .
Step 10.4.3.1.3.1.3
Rewrite as .
Step 10.4.3.1.3.1.4
Rewrite as .
Step 10.4.3.1.3.1.5
Multiply by .
Step 10.4.3.1.3.2
Subtract from .
Step 10.5
Move all terms not containing to the right side of the equation.
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Step 10.5.1
Subtract from both sides of the equation.
Step 10.5.2
Combine the opposite terms in .
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Step 10.5.2.1
Subtract from .
Step 10.5.2.2
Add and .
Step 11
Replace with to show the final answer.
Step 12
Verify if is the inverse of .
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Step 12.1
The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of and and compare them.
Step 12.2
Find the range of .
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Step 12.2.1
Find the range of .
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Step 12.2.1.1
The range is the set of all valid values. Use the graph to find the range.
Interval Notation:
Step 12.2.2
Find the range of .
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Step 12.2.2.1
The range is the set of all valid values. Use the graph to find the range.
Interval Notation:
Step 12.2.3
Find the union of .
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Step 12.2.3.1
The union consists of all of the elements that are contained in each interval.
Step 12.3
Find the domain of .
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Step 12.3.1
Set the radicand in greater than or equal to to find where the expression is defined.
Step 12.3.2
Subtract from both sides of the inequality.
Step 12.3.3
The domain is all values of that make the expression defined.
Step 12.4
Since the domain of is the range of and the range of is the domain of , then is the inverse of .
Step 13