Trigonometry Examples

Find the Inverse f(x)=6x^3-1
Step 1
Write as an equation.
Step 2
Interchange the variables.
Step 3
Solve for .
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Step 3.1
Rewrite the equation as .
Step 3.2
Add to both sides of the equation.
Step 3.3
Divide each term in by and simplify.
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Step 3.3.1
Divide each term in by .
Step 3.3.2
Simplify the left side.
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Step 3.3.2.1
Cancel the common factor of .
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Step 3.3.2.1.1
Cancel the common factor.
Step 3.3.2.1.2
Divide by .
Step 3.4
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 3.5
Simplify .
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Step 3.5.1
Combine the numerators over the common denominator.
Step 3.5.2
Rewrite as .
Step 3.5.3
Multiply by .
Step 3.5.4
Combine and simplify the denominator.
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Step 3.5.4.1
Multiply by .
Step 3.5.4.2
Raise to the power of .
Step 3.5.4.3
Use the power rule to combine exponents.
Step 3.5.4.4
Add and .
Step 3.5.4.5
Rewrite as .
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Step 3.5.4.5.1
Use to rewrite as .
Step 3.5.4.5.2
Apply the power rule and multiply exponents, .
Step 3.5.4.5.3
Combine and .
Step 3.5.4.5.4
Cancel the common factor of .
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Step 3.5.4.5.4.1
Cancel the common factor.
Step 3.5.4.5.4.2
Rewrite the expression.
Step 3.5.4.5.5
Evaluate the exponent.
Step 3.5.5
Simplify the numerator.
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Step 3.5.5.1
Rewrite as .
Step 3.5.5.2
Raise to the power of .
Step 3.5.6
Simplify with factoring out.
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Step 3.5.6.1
Combine using the product rule for radicals.
Step 3.5.6.2
Reorder factors in .
Step 4
Replace with to show the final answer.
Step 5
Verify if is the inverse of .
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Step 5.1
To verify the inverse, check if and .
Step 5.2
Evaluate .
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Step 5.2.1
Set up the composite result function.
Step 5.2.2
Evaluate by substituting in the value of into .
Step 5.2.3
Simplify the numerator.
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Step 5.2.3.1
Add and .
Step 5.2.3.2
Add and .
Step 5.2.3.3
Multiply by .
Step 5.2.3.4
Rewrite as .
Step 5.2.3.5
Pull terms out from under the radical, assuming real numbers.
Step 5.2.4
Cancel the common factor of .
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Step 5.2.4.1
Cancel the common factor.
Step 5.2.4.2
Divide by .
Step 5.3
Evaluate .
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Step 5.3.1
Set up the composite result function.
Step 5.3.2
Evaluate by substituting in the value of into .
Step 5.3.3
Simplify each term.
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Step 5.3.3.1
Apply the product rule to .
Step 5.3.3.2
Simplify the numerator.
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Step 5.3.3.2.1
Rewrite as .
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Step 5.3.3.2.1.1
Use to rewrite as .
Step 5.3.3.2.1.2
Apply the power rule and multiply exponents, .
Step 5.3.3.2.1.3
Combine and .
Step 5.3.3.2.1.4
Cancel the common factor of .
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Step 5.3.3.2.1.4.1
Cancel the common factor.
Step 5.3.3.2.1.4.2
Rewrite the expression.
Step 5.3.3.2.1.5
Simplify.
Step 5.3.3.2.2
Apply the distributive property.
Step 5.3.3.2.3
Multiply by .
Step 5.3.3.2.4
Factor out of .
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Step 5.3.3.2.4.1
Factor out of .
Step 5.3.3.2.4.2
Factor out of .
Step 5.3.3.2.4.3
Factor out of .
Step 5.3.3.3
Raise to the power of .
Step 5.3.3.4
Cancel the common factor of .
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Step 5.3.3.4.1
Factor out of .
Step 5.3.3.4.2
Cancel the common factor.
Step 5.3.3.4.3
Rewrite the expression.
Step 5.3.3.5
Cancel the common factor of .
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Step 5.3.3.5.1
Cancel the common factor.
Step 5.3.3.5.2
Divide by .
Step 5.3.4
Combine the opposite terms in .
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Step 5.3.4.1
Subtract from .
Step 5.3.4.2
Add and .
Step 5.4
Since and , then is the inverse of .