Trigonometry Examples

Find the Inverse y = square root of x-3
Step 1
Interchange the variables.
Step 2
Solve for .
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Step 2.1
Rewrite the equation as .
Step 2.2
Add to both sides of the equation.
Step 2.3
To remove the radical on the left side of the equation, square both sides of the equation.
Step 2.4
Simplify each side of the equation.
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Step 2.4.1
Use to rewrite as .
Step 2.4.2
Simplify the left side.
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Step 2.4.2.1
Simplify .
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Step 2.4.2.1.1
Multiply the exponents in .
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Step 2.4.2.1.1.1
Apply the power rule and multiply exponents, .
Step 2.4.2.1.1.2
Cancel the common factor of .
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Step 2.4.2.1.1.2.1
Cancel the common factor.
Step 2.4.2.1.1.2.2
Rewrite the expression.
Step 2.4.2.1.2
Simplify.
Step 2.4.3
Simplify the right side.
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Step 2.4.3.1
Simplify .
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Step 2.4.3.1.1
Rewrite as .
Step 2.4.3.1.2
Expand using the FOIL Method.
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Step 2.4.3.1.2.1
Apply the distributive property.
Step 2.4.3.1.2.2
Apply the distributive property.
Step 2.4.3.1.2.3
Apply the distributive property.
Step 2.4.3.1.3
Simplify and combine like terms.
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Step 2.4.3.1.3.1
Simplify each term.
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Step 2.4.3.1.3.1.1
Multiply by .
Step 2.4.3.1.3.1.2
Move to the left of .
Step 2.4.3.1.3.1.3
Multiply by .
Step 2.4.3.1.3.2
Add and .
Step 3
Replace with to show the final answer.
Step 4
Verify if is the inverse of .
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Step 4.1
To verify the inverse, check if and .
Step 4.2
Evaluate .
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Step 4.2.1
Set up the composite result function.
Step 4.2.2
Evaluate by substituting in the value of into .
Step 4.2.3
Simplify each term.
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Step 4.2.3.1
Rewrite as .
Step 4.2.3.2
Expand using the FOIL Method.
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Step 4.2.3.2.1
Apply the distributive property.
Step 4.2.3.2.2
Apply the distributive property.
Step 4.2.3.2.3
Apply the distributive property.
Step 4.2.3.3
Simplify and combine like terms.
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Step 4.2.3.3.1
Simplify each term.
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Step 4.2.3.3.1.1
Multiply .
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Step 4.2.3.3.1.1.1
Raise to the power of .
Step 4.2.3.3.1.1.2
Raise to the power of .
Step 4.2.3.3.1.1.3
Use the power rule to combine exponents.
Step 4.2.3.3.1.1.4
Add and .
Step 4.2.3.3.1.2
Rewrite as .
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Step 4.2.3.3.1.2.1
Use to rewrite as .
Step 4.2.3.3.1.2.2
Apply the power rule and multiply exponents, .
Step 4.2.3.3.1.2.3
Combine and .
Step 4.2.3.3.1.2.4
Cancel the common factor of .
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Step 4.2.3.3.1.2.4.1
Cancel the common factor.
Step 4.2.3.3.1.2.4.2
Rewrite the expression.
Step 4.2.3.3.1.2.5
Simplify.
Step 4.2.3.3.1.3
Move to the left of .
Step 4.2.3.3.1.4
Multiply by .
Step 4.2.3.3.2
Subtract from .
Step 4.2.3.4
Apply the distributive property.
Step 4.2.3.5
Multiply by .
Step 4.2.4
Simplify by adding terms.
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Step 4.2.4.1
Combine the opposite terms in .
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Step 4.2.4.1.1
Add and .
Step 4.2.4.1.2
Add and .
Step 4.2.4.2
Subtract from .
Step 4.2.4.3
Combine the opposite terms in .
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Step 4.2.4.3.1
Add and .
Step 4.2.4.3.2
Add and .
Step 4.3
Evaluate .
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Step 4.3.1
Set up the composite result function.
Step 4.3.2
Evaluate by substituting in the value of into .
Step 4.3.3
Remove parentheses.
Step 4.3.4
Simplify each term.
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Step 4.3.4.1
Factor using the perfect square rule.
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Step 4.3.4.1.1
Rewrite as .
Step 4.3.4.1.2
Check that the middle term is two times the product of the numbers being squared in the first term and third term.
Step 4.3.4.1.3
Rewrite the polynomial.
Step 4.3.4.1.4
Factor using the perfect square trinomial rule , where and .
Step 4.3.4.2
Pull terms out from under the radical, assuming positive real numbers.
Step 4.3.5
Combine the opposite terms in .
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Step 4.3.5.1
Subtract from .
Step 4.3.5.2
Add and .
Step 4.4
Since and , then is the inverse of .