Calculus Examples

Find the Inverse f(x)=(e^x)/(1+2e^x)
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
Multiply both sides by .
Step 3.3
Simplify.
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Step 3.3.1
Simplify the left side.
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Step 3.3.1.1
Cancel the common factor of .
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Step 3.3.1.1.1
Cancel the common factor.
Step 3.3.1.1.2
Rewrite the expression.
Step 3.3.2
Simplify the right side.
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Step 3.3.2.1
Simplify .
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Step 3.3.2.1.1
Apply the distributive property.
Step 3.3.2.1.2
Simplify the expression.
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Step 3.3.2.1.2.1
Multiply by .
Step 3.3.2.1.2.2
Rewrite using the commutative property of multiplication.
Step 3.4
Solve for .
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Step 3.4.1
Subtract from both sides of the equation.
Step 3.4.2
Factor out of .
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Step 3.4.2.1
Multiply by .
Step 3.4.2.2
Factor out of .
Step 3.4.2.3
Factor out of .
Step 3.4.3
Divide each term in by and simplify.
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Step 3.4.3.1
Divide each term in by .
Step 3.4.3.2
Simplify the left side.
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Step 3.4.3.2.1
Cancel the common factor of .
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Step 3.4.3.2.1.1
Cancel the common factor.
Step 3.4.3.2.1.2
Divide by .
Step 3.4.4
Take the natural logarithm of both sides of the equation to remove the variable from the exponent.
Step 3.4.5
Expand the left side.
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Step 3.4.5.1
Expand by moving outside the logarithm.
Step 3.4.5.2
The natural logarithm of is .
Step 3.4.5.3
Multiply by .
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
Multiply the numerator by the reciprocal of the denominator.
Step 5.2.4
Simplify the denominator.
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Step 5.2.4.1
Combine and .
Step 5.2.4.2
Move the negative in front of the fraction.
Step 5.2.4.3
Write as a fraction with a common denominator.
Step 5.2.4.4
Combine the numerators over the common denominator.
Step 5.2.4.5
Reorder terms.
Step 5.2.4.6
Rewrite in a factored form.
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Step 5.2.4.6.1
Subtract from .
Step 5.2.4.6.2
Add and .
Step 5.2.5
Multiply the numerator by the reciprocal of the denominator.
Step 5.2.6
Multiply by .
Step 5.2.7
Multiply by .
Step 5.2.8
Cancel the common factor of and .
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Step 5.2.8.1
Reorder terms.
Step 5.2.8.2
Cancel the common factor.
Step 5.2.8.3
Divide by .
Step 5.2.9
Use logarithm rules to move out of the exponent.
Step 5.2.10
The natural logarithm of is .
Step 5.2.11
Multiply 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
Exponentiation and log are inverse functions.
Step 5.3.4
Simplify the denominator.
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Step 5.3.4.1
Exponentiation and log are inverse functions.
Step 5.3.4.2
Combine and .
Step 5.3.4.3
Write as a fraction with a common denominator.
Step 5.3.4.4
Combine the numerators over the common denominator.
Step 5.3.4.5
Rewrite in a factored form.
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Step 5.3.4.5.1
Add and .
Step 5.3.4.5.2
Add and .
Step 5.3.5
Multiply the numerator by the reciprocal of the denominator.
Step 5.3.6
Cancel the common factor of .
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Step 5.3.6.1
Cancel the common factor.
Step 5.3.6.2
Rewrite the expression.
Step 5.4
Since and , then is the inverse of .