Calculus Examples

Find the Inverse f(x)=1+ square root of 2+3x
Step 1
Write as an equation.
Step 2
Interchange the variables.
Step 3
Solve for .
Tap for more steps...
Step 3.1
Rewrite the equation as .
Step 3.2
Subtract from both sides of the equation.
Step 3.3
To remove the radical on the left side of the equation, square both sides of the equation.
Step 3.4
Simplify each side of the equation.
Tap for more steps...
Step 3.4.1
Use to rewrite as .
Step 3.4.2
Simplify the left side.
Tap for more steps...
Step 3.4.2.1
Simplify .
Tap for more steps...
Step 3.4.2.1.1
Multiply the exponents in .
Tap for more steps...
Step 3.4.2.1.1.1
Apply the power rule and multiply exponents, .
Step 3.4.2.1.1.2
Cancel the common factor of .
Tap for more steps...
Step 3.4.2.1.1.2.1
Cancel the common factor.
Step 3.4.2.1.1.2.2
Rewrite the expression.
Step 3.4.2.1.2
Simplify.
Step 3.4.3
Simplify the right side.
Tap for more steps...
Step 3.4.3.1
Simplify .
Tap for more steps...
Step 3.4.3.1.1
Rewrite as .
Step 3.4.3.1.2
Expand using the FOIL Method.
Tap for more steps...
Step 3.4.3.1.2.1
Apply the distributive property.
Step 3.4.3.1.2.2
Apply the distributive property.
Step 3.4.3.1.2.3
Apply the distributive property.
Step 3.4.3.1.3
Simplify and combine like terms.
Tap for more steps...
Step 3.4.3.1.3.1
Simplify each term.
Tap for more steps...
Step 3.4.3.1.3.1.1
Multiply by .
Step 3.4.3.1.3.1.2
Move to the left of .
Step 3.4.3.1.3.1.3
Rewrite as .
Step 3.4.3.1.3.1.4
Rewrite as .
Step 3.4.3.1.3.1.5
Multiply by .
Step 3.4.3.1.3.2
Subtract from .
Step 3.5
Solve for .
Tap for more steps...
Step 3.5.1
Move all terms not containing to the right side of the equation.
Tap for more steps...
Step 3.5.1.1
Subtract from both sides of the equation.
Step 3.5.1.2
Subtract from .
Step 3.5.2
Divide each term in by and simplify.
Tap for more steps...
Step 3.5.2.1
Divide each term in by .
Step 3.5.2.2
Simplify the left side.
Tap for more steps...
Step 3.5.2.2.1
Cancel the common factor of .
Tap for more steps...
Step 3.5.2.2.1.1
Cancel the common factor.
Step 3.5.2.2.1.2
Divide by .
Step 3.5.2.3
Simplify the right side.
Tap for more steps...
Step 3.5.2.3.1
Simplify each term.
Tap for more steps...
Step 3.5.2.3.1.1
Move the negative in front of the fraction.
Step 3.5.2.3.1.2
Move the negative in front of the fraction.
Step 4
Replace with to show the final answer.
Step 5
Verify if is the inverse of .
Tap for more steps...
Step 5.1
To verify the inverse, check if and .
Step 5.2
Evaluate .
Tap for more steps...
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 numerators over the common denominator.
Step 5.2.4
Simplify each term.
Tap for more steps...
Step 5.2.4.1
Rewrite as .
Step 5.2.4.2
Expand using the FOIL Method.
Tap for more steps...
Step 5.2.4.2.1
Apply the distributive property.
Step 5.2.4.2.2
Apply the distributive property.
Step 5.2.4.2.3
Apply the distributive property.
Step 5.2.4.3
Simplify and combine like terms.
Tap for more steps...
Step 5.2.4.3.1
Simplify each term.
Tap for more steps...
Step 5.2.4.3.1.1
Multiply by .
Step 5.2.4.3.1.2
Multiply by .
Step 5.2.4.3.1.3
Multiply by .
Step 5.2.4.3.1.4
Multiply .
Tap for more steps...
Step 5.2.4.3.1.4.1
Raise to the power of .
Step 5.2.4.3.1.4.2
Raise to the power of .
Step 5.2.4.3.1.4.3
Use the power rule to combine exponents.
Step 5.2.4.3.1.4.4
Add and .
Step 5.2.4.3.1.5
Rewrite as .
Tap for more steps...
Step 5.2.4.3.1.5.1
Use to rewrite as .
Step 5.2.4.3.1.5.2
Apply the power rule and multiply exponents, .
Step 5.2.4.3.1.5.3
Combine and .
Step 5.2.4.3.1.5.4
Cancel the common factor of .
Tap for more steps...
Step 5.2.4.3.1.5.4.1
Cancel the common factor.
Step 5.2.4.3.1.5.4.2
Rewrite the expression.
Step 5.2.4.3.1.5.5
Simplify.
Step 5.2.4.3.2
Add and .
Step 5.2.4.3.3
Add and .
Step 5.2.4.4
Apply the distributive property.
Step 5.2.4.5
Multiply by .
Step 5.2.5
Simplify terms.
Tap for more steps...
Step 5.2.5.1
Combine the opposite terms in .
Tap for more steps...
Step 5.2.5.1.1
Subtract from .
Step 5.2.5.1.2
Add and .
Step 5.2.5.2
Subtract from .
Step 5.2.5.3
Combine the opposite terms in .
Tap for more steps...
Step 5.2.5.3.1
Subtract from .
Step 5.2.5.3.2
Add and .
Step 5.2.5.4
Cancel the common factor of .
Tap for more steps...
Step 5.2.5.4.1
Cancel the common factor.
Step 5.2.5.4.2
Divide by .
Step 5.3
Evaluate .
Tap for more steps...
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.
Tap for more steps...
Step 5.3.3.1
Apply the distributive property.
Step 5.3.3.2
Simplify.
Tap for more steps...
Step 5.3.3.2.1
Cancel the common factor of .
Tap for more steps...
Step 5.3.3.2.1.1
Cancel the common factor.
Step 5.3.3.2.1.2
Rewrite the expression.
Step 5.3.3.2.2
Cancel the common factor of .
Tap for more steps...
Step 5.3.3.2.2.1
Move the leading negative in into the numerator.
Step 5.3.3.2.2.2
Cancel the common factor.
Step 5.3.3.2.2.3
Rewrite the expression.
Step 5.3.3.2.3
Cancel the common factor of .
Tap for more steps...
Step 5.3.3.2.3.1
Move the leading negative in into the numerator.
Step 5.3.3.2.3.2
Cancel the common factor.
Step 5.3.3.2.3.3
Rewrite the expression.
Step 5.3.3.3
Subtract from .
Step 5.3.3.4
Factor using the perfect square rule.
Tap for more steps...
Step 5.3.3.4.1
Rewrite as .
Step 5.3.3.4.2
Check that the middle term is two times the product of the numbers being squared in the first term and third term.
Step 5.3.3.4.3
Rewrite the polynomial.
Step 5.3.3.4.4
Factor using the perfect square trinomial rule , where and .
Step 5.3.3.5
Pull terms out from under the radical, assuming positive real numbers.
Step 5.3.4
Combine the opposite terms in .
Tap for more steps...
Step 5.3.4.1
Subtract from .
Step 5.3.4.2
Add and .
Step 5.4
Since and , then is the inverse of .