Precalculus Examples

$f (x) = x^{\frac{1}{3}} + 1$

Step 1

Write $f (x) = x^{\frac{1}{3}} + 1$ as an equation.

$y = x^{\frac{1}{3}} + 1$

Step 2

Interchange the variables.

$x = y^{\frac{1}{3}} + 1$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $y^{\frac{1}{3}} + 1 = x$ .

$y^{\frac{1}{3}} + 1 = x$

Step 3.2

Move all terms not containing $y$ to the right side of the equation.

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Step 3.2.1

Subtract $1$ from both sides of the equation.

$y^{\frac{1}{3}} - x = - 1$

Step 3.2.2

Add $x$ to both sides of the equation.

$y^{\frac{1}{3}} = - 1 + x$

Step 3.3

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${(y^{\frac{1}{3}})}^{3} = {(- 1 + x)}^{3}$

Step 3.4

Simplify the exponent.

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Step 3.4.1

Simplify the left side.

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Step 3.4.1.1

Simplify ${(y^{\frac{1}{3}})}^{3}$ .

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Step 3.4.1.1.1

Multiply the exponents in ${(y^{\frac{1}{3}})}^{3}$ .

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Step 3.4.1.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y^{\frac{1}{3} \cdot 3} = {(- 1 + x)}^{3}$

Step 3.4.1.1.1.2

Cancel the common factor of $3$ .

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Step 3.4.1.1.1.2.1

Cancel the common factor.

$y^{\frac{1}{3} \cdot 3} = {(- 1 + x)}^{3}$

Step 3.4.1.1.1.2.2

Rewrite the expression.

$y^{1} = {(- 1 + x)}^{3}$

Step 3.4.1.1.2

Simplify.

$y = {(- 1 + x)}^{3}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify ${(- 1 + x)}^{3}$ .

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Step 3.4.2.1.1

Use the Binomial Theorem.

$y = {(- 1)}^{3} + 3 {(- 1)}^{2} x + 3 \cdot - 1 x^{2} + x^{3}$

Step 3.4.2.1.2

Simplify each term.

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Step 3.4.2.1.2.1

Raise $- 1$ to the power of $3$ .

$y = - 1 + 3 {(- 1)}^{2} x + 3 \cdot - 1 x^{2} + x^{3}$

Step 3.4.2.1.2.2

Raise $- 1$ to the power of $2$ .

$y = - 1 + 3 \cdot 1 x + 3 \cdot - 1 x^{2} + x^{3}$

Step 3.4.2.1.2.3

Multiply $3$ by $1$ .

$y = - 1 + 3 x + 3 \cdot - 1 x^{2} + x^{3}$

Step 3.4.2.1.2.4

Multiply $3$ by $- 1$ .

$y = - 1 + 3 x - 3 x^{2} + x^{3}$

Step 3.5

Simplify $- 1 + 3 x - 3 x^{2} + x^{3}$ .

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Step 3.5.1

Move $- 1$ .

$y = 3 x - 3 x^{2} + x^{3} - 1$

Step 3.5.2

Move $3 x$ .

$y = - 3 x^{2} + x^{3} + 3 x - 1$

Step 3.5.3

Reorder $- 3 x^{2}$ and $x^{3}$ .

$y = x^{3} - 3 x^{2} + 3 x - 1$

Step 4

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = x^{3} - 3 x^{2} + 3 x - 1$

Step 5

Verify if $f^{- 1} (x) = x^{3} - 3 x^{2} + 3 x - 1$ is the inverse of $f (x) = x^{\frac{1}{3}} + 1$ .

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Step 5.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 5.2

Evaluate $f^{- 1} (f (x))$ .

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Step 5.2.1

Set up the composite result function.

$f^{- 1} (f (x))$

Step 5.2.2

Evaluate $f^{- 1} (x^{\frac{1}{3}} + 1)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (x^{\frac{1}{3}} + 1) = {(x^{\frac{1}{3}} + 1)}^{3} - 3 {(x^{\frac{1}{3}} + 1)}^{2} + 3 (x^{\frac{1}{3}} + 1) - 1$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Use the Binomial Theorem.

$f^{- 1} (x^{\frac{1}{3}} + 1) = {(x^{\frac{1}{3}})}^{3} + 3 {(x^{\frac{1}{3}})}^{2} \cdot 1 + 3 x^{\frac{1}{3}} \cdot 1^{2} + 1^{3} - 3 {(x^{\frac{1}{3}} + 1)}^{2} + 3 (x^{\frac{1}{3}} + 1) - 1$

Step 5.2.3.2

Simplify each term.

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Step 5.2.3.2.1

Multiply the exponents in ${(x^{\frac{1}{3}})}^{3}$ .

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Step 5.2.3.2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f^{- 1} (x^{\frac{1}{3}} + 1) = x^{\frac{1}{3} \cdot 3} + 3 {(x^{\frac{1}{3}})}^{2} \cdot 1 + 3 x^{\frac{1}{3}} \cdot 1^{2} + 1^{3} - 3 {(x^{\frac{1}{3}} + 1)}^{2} + 3 (x^{\frac{1}{3}} + 1) - 1$

Step 5.2.3.2.1.2

Cancel the common factor of $3$ .

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Step 5.2.3.2.1.2.1

Cancel the common factor.

$f^{- 1} (x^{\frac{1}{3}} + 1) = x^{\frac{1}{3} \cdot 3} + 3 {(x^{\frac{1}{3}})}^{2} \cdot 1 + 3 x^{\frac{1}{3}} \cdot 1^{2} + 1^{3} - 3 {(x^{\frac{1}{3}} + 1)}^{2} + 3 (x^{\frac{1}{3}} + 1) - 1$

Step 5.2.3.2.1.2.2

Rewrite the expression.

$f^{- 1} (x^{\frac{1}{3}} + 1) = x + 3 {(x^{\frac{1}{3}})}^{2} \cdot 1 + 3 x^{\frac{1}{3}} \cdot 1^{2} + 1^{3} - 3 {(x^{\frac{1}{3}} + 1)}^{2} + 3 (x^{\frac{1}{3}} + 1) - 1$

Step 5.2.3.2.2

Simplify.

$f^{- 1} (x^{\frac{1}{3}} + 1) = x + 3 {(x^{\frac{1}{3}})}^{2} \cdot 1 + 3 x^{\frac{1}{3}} \cdot 1^{2} + 1^{3} - 3 {(x^{\frac{1}{3}} + 1)}^{2} + 3 (x^{\frac{1}{3}} + 1) - 1$

Step 5.2.3.2.3

Multiply the exponents in ${(x^{\frac{1}{3}})}^{2}$ .

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Step 5.2.3.2.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f^{- 1} (x^{\frac{1}{3}} + 1) = x + 3 x^{\frac{1}{3} \cdot 2} \cdot 1 + 3 x^{\frac{1}{3}} \cdot 1^{2} + 1^{3} - 3 {(x^{\frac{1}{3}} + 1)}^{2} + 3 (x^{\frac{1}{3}} + 1) - 1$

Step 5.2.3.2.3.2

Combine $\frac{1}{3}$ and $2$ .

$f^{- 1} (x^{\frac{1}{3}} + 1) = x + 3 x^{\frac{2}{3}} \cdot 1 + 3 x^{\frac{1}{3}} \cdot 1^{2} + 1^{3} - 3 {(x^{\frac{1}{3}} + 1)}^{2} + 3 (x^{\frac{1}{3}} + 1) - 1$

Step 5.2.3.2.4

Multiply $3$ by $1$ .

$f^{- 1} (x^{\frac{1}{3}} + 1) = x + 3 x^{\frac{2}{3}} + 3 x^{\frac{1}{3}} \cdot 1^{2} + 1^{3} - 3 {(x^{\frac{1}{3}} + 1)}^{2} + 3 (x^{\frac{1}{3}} + 1) - 1$

Step 5.2.3.2.5

One to any power is one.

$f^{- 1} (x^{\frac{1}{3}} + 1) = x + 3 x^{\frac{2}{3}} + 3 x^{\frac{1}{3}} \cdot 1 + 1^{3} - 3 {(x^{\frac{1}{3}} + 1)}^{2} + 3 (x^{\frac{1}{3}} + 1) - 1$

Step 5.2.3.2.6

Multiply $3$ by $1$ .

$f^{- 1} (x^{\frac{1}{3}} + 1) = x + 3 x^{\frac{2}{3}} + 3 x^{\frac{1}{3}} + 1^{3} - 3 {(x^{\frac{1}{3}} + 1)}^{2} + 3 (x^{\frac{1}{3}} + 1) - 1$

Step 5.2.3.2.7

One to any power is one.

$f^{- 1} (x^{\frac{1}{3}} + 1) = x + 3 x^{\frac{2}{3}} + 3 x^{\frac{1}{3}} + 1 - 3 {(x^{\frac{1}{3}} + 1)}^{2} + 3 (x^{\frac{1}{3}} + 1) - 1$

Step 5.2.3.3

Rewrite ${(x^{\frac{1}{3}} + 1)}^{2}$ as $(x^{\frac{1}{3}} + 1) (x^{\frac{1}{3}} + 1)$ .

$f^{- 1} (x^{\frac{1}{3}} + 1) = x + 3 x^{\frac{2}{3}} + 3 x^{\frac{1}{3}} + 1 - 3 ((x^{\frac{1}{3}} + 1) (x^{\frac{1}{3}} + 1)) + 3 (x^{\frac{1}{3}} + 1) - 1$

Step 5.2.3.4

Expand $(x^{\frac{1}{3}} + 1) (x^{\frac{1}{3}} + 1)$ using the FOIL Method.

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Step 5.2.3.4.1

Apply the distributive property.

$f^{- 1} (x^{\frac{1}{3}} + 1) = x + 3 x^{\frac{2}{3}} + 3 x^{\frac{1}{3}} + 1 - 3 (x^{\frac{1}{3}} (x^{\frac{1}{3}} + 1) + 1 (x^{\frac{1}{3}} + 1)) + 3 (x^{\frac{1}{3}} + 1) - 1$

Step 5.2.3.4.2

Apply the distributive property.

$f^{- 1} (x^{\frac{1}{3}} + 1) = x + 3 x^{\frac{2}{3}} + 3 x^{\frac{1}{3}} + 1 - 3 (x^{\frac{1}{3}} x^{\frac{1}{3}} + x^{\frac{1}{3}} \cdot 1 + 1 (x^{\frac{1}{3}} + 1)) + 3 (x^{\frac{1}{3}} + 1) - 1$

Step 5.2.3.4.3

Apply the distributive property.

$f^{- 1} (x^{\frac{1}{3}} + 1) = x + 3 x^{\frac{2}{3}} + 3 x^{\frac{1}{3}} + 1 - 3 (x^{\frac{1}{3}} x^{\frac{1}{3}} + x^{\frac{1}{3}} \cdot 1 + 1 x^{\frac{1}{3}} + 1 \cdot 1) + 3 (x^{\frac{1}{3}} + 1) - 1$

Step 5.2.3.5

Simplify and combine like terms.

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Step 5.2.3.5.1

Simplify each term.

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Step 5.2.3.5.1.1

Multiply $x^{\frac{1}{3}}$ by $x^{\frac{1}{3}}$ by adding the exponents.

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Step 5.2.3.5.1.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f^{- 1} (x^{\frac{1}{3}} + 1) = x + 3 x^{\frac{2}{3}} + 3 x^{\frac{1}{3}} + 1 - 3 (x^{\frac{1}{3} + \frac{1}{3}} + x^{\frac{1}{3}} \cdot 1 + 1 x^{\frac{1}{3}} + 1 \cdot 1) + 3 (x^{\frac{1}{3}} + 1) - 1$

Step 5.2.3.5.1.1.2

Combine the numerators over the common denominator.

$f^{- 1} (x^{\frac{1}{3}} + 1) = x + 3 x^{\frac{2}{3}} + 3 x^{\frac{1}{3}} + 1 - 3 (x^{\frac{1 + 1}{3}} + x^{\frac{1}{3}} \cdot 1 + 1 x^{\frac{1}{3}} + 1 \cdot 1) + 3 (x^{\frac{1}{3}} + 1) - 1$

Step 5.2.3.5.1.1.3

Add $1$ and $1$ .

$f^{- 1} (x^{\frac{1}{3}} + 1) = x + 3 x^{\frac{2}{3}} + 3 x^{\frac{1}{3}} + 1 - 3 (x^{\frac{2}{3}} + x^{\frac{1}{3}} \cdot 1 + 1 x^{\frac{1}{3}} + 1 \cdot 1) + 3 (x^{\frac{1}{3}} + 1) - 1$

Step 5.2.3.5.1.2

Multiply $x^{\frac{1}{3}}$ by $1$ .

$f^{- 1} (x^{\frac{1}{3}} + 1) = x + 3 x^{\frac{2}{3}} + 3 x^{\frac{1}{3}} + 1 - 3 (x^{\frac{2}{3}} + x^{\frac{1}{3}} + 1 x^{\frac{1}{3}} + 1 \cdot 1) + 3 (x^{\frac{1}{3}} + 1) - 1$

Step 5.2.3.5.1.3

Multiply $x^{\frac{1}{3}}$ by $1$ .

$f^{- 1} (x^{\frac{1}{3}} + 1) = x + 3 x^{\frac{2}{3}} + 3 x^{\frac{1}{3}} + 1 - 3 (x^{\frac{2}{3}} + x^{\frac{1}{3}} + x^{\frac{1}{3}} + 1 \cdot 1) + 3 (x^{\frac{1}{3}} + 1) - 1$

Step 5.2.3.5.1.4

Multiply $1$ by $1$ .

$f^{- 1} (x^{\frac{1}{3}} + 1) = x + 3 x^{\frac{2}{3}} + 3 x^{\frac{1}{3}} + 1 - 3 (x^{\frac{2}{3}} + x^{\frac{1}{3}} + x^{\frac{1}{3}} + 1) + 3 (x^{\frac{1}{3}} + 1) - 1$

Step 5.2.3.5.2

Add $x^{\frac{1}{3}}$ and $x^{\frac{1}{3}}$ .

$f^{- 1} (x^{\frac{1}{3}} + 1) = x + 3 x^{\frac{2}{3}} + 3 x^{\frac{1}{3}} + 1 - 3 (x^{\frac{2}{3}} + 2 x^{\frac{1}{3}} + 1) + 3 (x^{\frac{1}{3}} + 1) - 1$

Step 5.2.3.6

Apply the distributive property.

$f^{- 1} (x^{\frac{1}{3}} + 1) = x + 3 x^{\frac{2}{3}} + 3 x^{\frac{1}{3}} + 1 - 3 x^{\frac{2}{3}} - 3 (2 x^{\frac{1}{3}}) - 3 \cdot 1 + 3 (x^{\frac{1}{3}} + 1) - 1$

Step 5.2.3.7

Simplify.

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Step 5.2.3.7.1

Multiply $2$ by $- 3$ .

$f^{- 1} (x^{\frac{1}{3}} + 1) = x + 3 x^{\frac{2}{3}} + 3 x^{\frac{1}{3}} + 1 - 3 x^{\frac{2}{3}} - 6 x^{\frac{1}{3}} - 3 \cdot 1 + 3 (x^{\frac{1}{3}} + 1) - 1$

Step 5.2.3.7.2

Multiply $- 3$ by $1$ .

$f^{- 1} (x^{\frac{1}{3}} + 1) = x + 3 x^{\frac{2}{3}} + 3 x^{\frac{1}{3}} + 1 - 3 x^{\frac{2}{3}} - 6 x^{\frac{1}{3}} - 3 + 3 (x^{\frac{1}{3}} + 1) - 1$

Step 5.2.3.8

Apply the distributive property.

$f^{- 1} (x^{\frac{1}{3}} + 1) = x + 3 x^{\frac{2}{3}} + 3 x^{\frac{1}{3}} + 1 - 3 x^{\frac{2}{3}} - 6 x^{\frac{1}{3}} - 3 + 3 x^{\frac{1}{3}} + 3 \cdot 1 - 1$

Step 5.2.3.9

Multiply $3$ by $1$ .

$f^{- 1} (x^{\frac{1}{3}} + 1) = x + 3 x^{\frac{2}{3}} + 3 x^{\frac{1}{3}} + 1 - 3 x^{\frac{2}{3}} - 6 x^{\frac{1}{3}} - 3 + 3 x^{\frac{1}{3}} + 3 - 1$

Step 5.2.4

Simplify by adding terms.

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Step 5.2.4.1

Combine the opposite terms in $x + 3 x^{\frac{2}{3}} + 3 x^{\frac{1}{3}} + 1 - 3 x^{\frac{2}{3}} - 6 x^{\frac{1}{3}} - 3 + 3 x^{\frac{1}{3}} + 3 - 1$ .

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Step 5.2.4.1.1

Subtract $3 x^{\frac{2}{3}}$ from $3 x^{\frac{2}{3}}$ .

$f^{- 1} (x^{\frac{1}{3}} + 1) = x + 3 x^{\frac{1}{3}} + 1 + 0 - 6 x^{\frac{1}{3}} - 3 + 3 x^{\frac{1}{3}} + 3 - 1$

Step 5.2.4.1.2

Add $x + 3 x^{\frac{1}{3}} + 1$ and $0$ .

$f^{- 1} (x^{\frac{1}{3}} + 1) = x + 3 x^{\frac{1}{3}} + 1 - 6 x^{\frac{1}{3}} - 3 + 3 x^{\frac{1}{3}} + 3 - 1$

Step 5.2.4.1.3

Add $- 3$ and $3$ .

$f^{- 1} (x^{\frac{1}{3}} + 1) = x + 3 x^{\frac{1}{3}} + 1 - 6 x^{\frac{1}{3}} + 3 x^{\frac{1}{3}} + 0 - 1$

Step 5.2.4.1.4

Add $x + 3 x^{\frac{1}{3}} + 1 - 6 x^{\frac{1}{3}} + 3 x^{\frac{1}{3}}$ and $0$ .

$f^{- 1} (x^{\frac{1}{3}} + 1) = x + 3 x^{\frac{1}{3}} + 1 - 6 x^{\frac{1}{3}} + 3 x^{\frac{1}{3}} - 1$

Step 5.2.4.1.5

Subtract $1$ from $1$ .

$f^{- 1} (x^{\frac{1}{3}} + 1) = x + 3 x^{\frac{1}{3}} - 6 x^{\frac{1}{3}} + 3 x^{\frac{1}{3}} + 0$

Step 5.2.4.1.6

Add $x + 3 x^{\frac{1}{3}} - 6 x^{\frac{1}{3}} + 3 x^{\frac{1}{3}}$ and $0$ .

$f^{- 1} (x^{\frac{1}{3}} + 1) = x + 3 x^{\frac{1}{3}} - 6 x^{\frac{1}{3}} + 3 x^{\frac{1}{3}}$

Step 5.2.4.2

Subtract $6 x^{\frac{1}{3}}$ from $3 x^{\frac{1}{3}}$ .

$f^{- 1} (x^{\frac{1}{3}} + 1) = x - 3 x^{\frac{1}{3}} + 3 x^{\frac{1}{3}}$

Step 5.2.4.3

Combine the opposite terms in $x - 3 x^{\frac{1}{3}} + 3 x^{\frac{1}{3}}$ .

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Step 5.2.4.3.1

Add $- 3 x^{\frac{1}{3}}$ and $3 x^{\frac{1}{3}}$ .

$f^{- 1} (x^{\frac{1}{3}} + 1) = x + 0$

Step 5.2.4.3.2

Add $x$ and $0$ .

$f^{- 1} (x^{\frac{1}{3}} + 1) = x$

Step 5.3

Evaluate $f (f^{- 1} (x))$ .

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Step 5.3.1

Set up the composite result function.

$f (f^{- 1} (x))$

Step 5.3.2

Evaluate $f (x^{3} - 3 x^{2} + 3 x - 1)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (x^{3} - 3 x^{2} + 3 x - 1) = {(x^{3} - 3 x^{2} + 3 x - 1)}^{\frac{1}{3}} + 1$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = x^{3} - 3 x^{2} + 3 x - 1$ is the inverse of $f (x) = x^{\frac{1}{3}} + 1$ .

$f^{- 1} (x) = x^{3} - 3 x^{2} + 3 x - 1$