Finite Math Examples

Find the Inverse f(x)=(7x+4)/(x-5)
Step 1
Write as an equation.
Step 2
Interchange the variables.
Step 3
Solve for .
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Step 3.1
Multiply the equation by .
Step 3.2
Simplify the left side.
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Step 3.2.1
Apply the distributive property.
Step 3.3
Simplify the right side.
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Step 3.3.1
Cancel the common factor of .
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Step 3.3.1.1
Cancel the common factor.
Step 3.3.1.2
Rewrite the expression.
Step 3.4
Solve for .
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Step 3.4.1
Subtract from both sides of the equation.
Step 3.4.2
Add to both sides of the equation.
Step 3.4.3
Factor out of .
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Step 3.4.3.1
Factor out of .
Step 3.4.3.2
Factor out of .
Step 3.4.3.3
Factor out of .
Step 3.4.4
Divide each term in by and simplify.
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Step 3.4.4.1
Divide each term in by .
Step 3.4.4.2
Simplify the left side.
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Step 3.4.4.2.1
Cancel the common factor of .
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Step 3.4.4.2.1.1
Cancel the common factor.
Step 3.4.4.2.1.2
Divide by .
Step 3.4.4.3
Simplify the right side.
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Step 3.4.4.3.1
Combine the numerators over the common denominator.
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 and denominator of the fraction by .
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Step 5.2.3.1
Multiply by .
Step 5.2.3.2
Combine.
Step 5.2.4
Apply the distributive property.
Step 5.2.5
Cancel the common factor of .
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Step 5.2.5.1
Cancel the common factor.
Step 5.2.5.2
Rewrite the expression.
Step 5.2.6
Simplify the numerator.
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Step 5.2.6.1
Factor out of .
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Step 5.2.6.1.1
Factor out of .
Step 5.2.6.1.2
Factor out of .
Step 5.2.6.2
Combine and .
Step 5.2.6.3
To write as a fraction with a common denominator, multiply by .
Step 5.2.6.4
Combine the numerators over the common denominator.
Step 5.2.6.5
Reorder terms.
Step 5.2.6.6
Rewrite in a factored form.
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Step 5.2.6.6.1
Apply the distributive property.
Step 5.2.6.6.2
Multiply by .
Step 5.2.6.6.3
Multiply by .
Step 5.2.6.6.4
Apply the distributive property.
Step 5.2.6.6.5
Multiply by .
Step 5.2.6.6.6
Add and .
Step 5.2.6.6.7
Subtract from .
Step 5.2.6.6.8
Add and .
Step 5.2.7
Simplify the denominator.
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Step 5.2.7.1
Apply the distributive property.
Step 5.2.7.2
Move to the left of .
Step 5.2.7.3
Multiply by .
Step 5.2.7.4
Subtract from .
Step 5.2.7.5
Add and .
Step 5.2.7.6
Add and .
Step 5.2.8
Reduce the expression by cancelling the common factors.
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Step 5.2.8.1
Factor out of .
Step 5.2.8.2
Cancel the common factor of .
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Step 5.2.8.2.1
Factor out of .
Step 5.2.8.2.2
Cancel the common factor.
Step 5.2.8.2.3
Rewrite the expression.
Step 5.2.8.3
Cancel the common factor of .
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Step 5.2.8.3.1
Cancel the common factor.
Step 5.2.8.3.2
Rewrite the expression.
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
Multiply the numerator and denominator of the fraction by .
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Step 5.3.3.1
Multiply by .
Step 5.3.3.2
Combine.
Step 5.3.4
Apply the distributive property.
Step 5.3.5
Cancel the common factor of .
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Step 5.3.5.1
Cancel the common factor.
Step 5.3.5.2
Rewrite the expression.
Step 5.3.6
Simplify the numerator.
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Step 5.3.6.1
Factor out of .
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Step 5.3.6.1.1
Factor out of .
Step 5.3.6.1.2
Factor out of .
Step 5.3.6.1.3
Factor out of .
Step 5.3.6.2
Combine and .
Step 5.3.6.3
To write as a fraction with a common denominator, multiply by .
Step 5.3.6.4
Combine the numerators over the common denominator.
Step 5.3.6.5
Rewrite in a factored form.
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Step 5.3.6.5.1
Apply the distributive property.
Step 5.3.6.5.2
Multiply by .
Step 5.3.6.5.3
Multiply by .
Step 5.3.6.5.4
Apply the distributive property.
Step 5.3.6.5.5
Multiply by .
Step 5.3.6.5.6
Subtract from .
Step 5.3.6.5.7
Add and .
Step 5.3.6.5.8
Add and .
Step 5.3.7
Simplify the denominator.
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Step 5.3.7.1
Apply the distributive property.
Step 5.3.7.2
Move to the left of .
Step 5.3.7.3
Multiply by .
Step 5.3.7.4
Add and .
Step 5.3.7.5
Subtract from .
Step 5.3.7.6
Add and .
Step 5.3.8
Reduce the expression by cancelling the common factors.
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Step 5.3.8.1
Factor out of .
Step 5.3.8.2
Cancel the common factor of .
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Step 5.3.8.2.1
Factor out of .
Step 5.3.8.2.2
Cancel the common factor.
Step 5.3.8.2.3
Rewrite the expression.
Step 5.3.8.3
Cancel the common factor of .
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Step 5.3.8.3.1
Cancel the common factor.
Step 5.3.8.3.2
Rewrite the expression.
Step 5.4
Since and , then is the inverse of .