Trigonometry Examples

Find the Inverse f(x)=2^(x-1)+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
Subtract from both sides of the equation.
Step 3.3
Take the natural logarithm of both sides of the equation to remove the variable from the exponent.
Step 3.4
Expand by moving outside the logarithm.
Step 3.5
Simplify the left side.
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Step 3.5.1
Simplify .
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Step 3.5.1.1
Apply the distributive property.
Step 3.5.1.2
Rewrite as .
Step 3.6
Move all the terms containing a logarithm to the left side of the equation.
Step 3.7
Move all terms not containing to the right side of the equation.
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Step 3.7.1
Add to both sides of the equation.
Step 3.7.2
Add to both sides of the equation.
Step 3.8
Divide each term in by and simplify.
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Step 3.8.1
Divide each term in by .
Step 3.8.2
Simplify the left side.
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Step 3.8.2.1
Cancel the common factor of .
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Step 3.8.2.1.1
Cancel the common factor.
Step 3.8.2.1.2
Divide by .
Step 3.8.3
Simplify the right side.
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Step 3.8.3.1
Cancel the common factor of .
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Step 3.8.3.1.1
Cancel the common factor.
Step 3.8.3.1.2
Rewrite the expression.
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
Combine the opposite terms in .
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Step 5.2.3.1
Subtract from .
Step 5.2.3.2
Add and .
Step 5.2.4
Simplify each term.
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Step 5.2.4.1
Expand by moving outside the logarithm.
Step 5.2.4.2
Cancel the common factor of .
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Step 5.2.4.2.1
Cancel the common factor.
Step 5.2.4.2.2
Divide by .
Step 5.2.5
Combine the opposite terms in .
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Step 5.2.5.1
Subtract from .
Step 5.2.5.2
Add and .
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
Combine the opposite terms in .
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Step 5.3.3.1
Subtract from .
Step 5.3.3.2
Add and .
Step 5.3.4
Simplify each term.
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Step 5.3.4.1
Use the change of base rule .
Step 5.3.4.2
Exponentiation and log are inverse functions.
Step 5.3.5
Combine the opposite terms in .
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Step 5.3.5.1
Add and .
Step 5.3.5.2
Add and .
Step 5.4
Since and , then is the inverse of .