Algebra Examples

$y = \sqrt[3]{3 x - 1} + 3$

Step 1

Interchange the variables.

$x = \sqrt[3]{3 y - 1} + 3$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\sqrt[3]{3 y - 1} + 3 = x$ .

$\sqrt[3]{3 y - 1} + 3 = x$

Step 2.2

Subtract $3$ from both sides of the equation.

$\sqrt[3]{3 y - 1} = x - 3$

Step 2.3

To remove the radical on the left side of the equation, cube both sides of the equation.

${\sqrt[3]{3 y - 1}}^{3} = {(x - 3)}^{3}$

Step 2.4

Simplify each side of the equation.

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Step 2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{3 y - 1}$ as ${(3 y - 1)}^{\frac{1}{3}}$ .

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

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

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

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Step 2.4.2.1.1

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

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Step 2.4.2.1.1.1

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

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

Step 2.4.2.1.1.2

Cancel the common factor of $3$ .

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Step 2.4.2.1.1.2.1

Cancel the common factor.

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

Step 2.4.2.1.1.2.2

Rewrite the expression.

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

Step 2.4.2.1.2

Simplify.

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

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

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

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Step 2.4.3.1.1

Use the Binomial Theorem.

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

Step 2.4.3.1.2

Simplify each term.

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Step 2.4.3.1.2.1

Multiply $- 3$ by $3$ .

$3 y - 1 = x^{3} - 9 x^{2} + 3 x {(- 3)}^{2} + {(- 3)}^{3}$

Step 2.4.3.1.2.2

Raise $- 3$ to the power of $2$ .

$3 y - 1 = x^{3} - 9 x^{2} + 3 x \cdot 9 + {(- 3)}^{3}$

Step 2.4.3.1.2.3

Multiply $9$ by $3$ .

$3 y - 1 = x^{3} - 9 x^{2} + 27 x + {(- 3)}^{3}$

Step 2.4.3.1.2.4

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

$3 y - 1 = x^{3} - 9 x^{2} + 27 x - 27$

Step 2.5

Solve for $y$ .

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Step 2.5.1

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

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Step 2.5.1.1

Add $1$ to both sides of the equation.

$3 y = x^{3} - 9 x^{2} + 27 x - 27 + 1$

Step 2.5.1.2

Add $- 27$ and $1$ .

$3 y = x^{3} - 9 x^{2} + 27 x - 26$

Step 2.5.2

Divide each term in $3 y = x^{3} - 9 x^{2} + 27 x - 26$ by $3$ and simplify.

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Step 2.5.2.1

Divide each term in $3 y = x^{3} - 9 x^{2} + 27 x - 26$ by $3$ .

$\frac{3 y}{3} = \frac{x^{3}}{3} + \frac{- 9 x^{2}}{3} + \frac{27 x}{3} + \frac{- 26}{3}$

Step 2.5.2.2

Simplify the left side.

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Step 2.5.2.2.1

Cancel the common factor of $3$ .

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Step 2.5.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{x^{3}}{3} + \frac{- 9 x^{2}}{3} + \frac{27 x}{3} + \frac{- 26}{3}$

Step 2.5.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{x^{3}}{3} + \frac{- 9 x^{2}}{3} + \frac{27 x}{3} + \frac{- 26}{3}$

Step 2.5.2.3

Simplify the right side.

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Step 2.5.2.3.1

Simplify each term.

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Step 2.5.2.3.1.1

Cancel the common factor of $- 9$ and $3$ .

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Step 2.5.2.3.1.1.1

Factor $3$ out of $- 9 x^{2}$ .

$y = \frac{x^{3}}{3} + \frac{3 (- 3 x^{2})}{3} + \frac{27 x}{3} + \frac{- 26}{3}$

Step 2.5.2.3.1.1.2

Cancel the common factors.

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Step 2.5.2.3.1.1.2.1

Factor $3$ out of $3$ .

$y = \frac{x^{3}}{3} + \frac{3 (- 3 x^{2})}{3 (1)} + \frac{27 x}{3} + \frac{- 26}{3}$

Step 2.5.2.3.1.1.2.2

Cancel the common factor.

$y = \frac{x^{3}}{3} + \frac{3 (- 3 x^{2})}{3 \cdot 1} + \frac{27 x}{3} + \frac{- 26}{3}$

Step 2.5.2.3.1.1.2.3

Rewrite the expression.

$y = \frac{x^{3}}{3} + \frac{- 3 x^{2}}{1} + \frac{27 x}{3} + \frac{- 26}{3}$

Step 2.5.2.3.1.1.2.4

Divide $- 3 x^{2}$ by $1$ .

$y = \frac{x^{3}}{3} - 3 x^{2} + \frac{27 x}{3} + \frac{- 26}{3}$

Step 2.5.2.3.1.2

Cancel the common factor of $27$ and $3$ .

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Step 2.5.2.3.1.2.1

Factor $3$ out of $27 x$ .

$y = \frac{x^{3}}{3} - 3 x^{2} + \frac{3 (9 x)}{3} + \frac{- 26}{3}$

Step 2.5.2.3.1.2.2

Cancel the common factors.

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Step 2.5.2.3.1.2.2.1

Factor $3$ out of $3$ .

$y = \frac{x^{3}}{3} - 3 x^{2} + \frac{3 (9 x)}{3 (1)} + \frac{- 26}{3}$

Step 2.5.2.3.1.2.2.2

Cancel the common factor.

$y = \frac{x^{3}}{3} - 3 x^{2} + \frac{3 (9 x)}{3 \cdot 1} + \frac{- 26}{3}$

Step 2.5.2.3.1.2.2.3

Rewrite the expression.

$y = \frac{x^{3}}{3} - 3 x^{2} + \frac{9 x}{1} + \frac{- 26}{3}$

Step 2.5.2.3.1.2.2.4

Divide $9 x$ by $1$ .

$y = \frac{x^{3}}{3} - 3 x^{2} + 9 x + \frac{- 26}{3}$

Step 2.5.2.3.1.3

Move the negative in front of the fraction.

$y = \frac{x^{3}}{3} - 3 x^{2} + 9 x - \frac{26}{3}$

Step 3

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

$f^{- 1} (x) = \frac{x^{3}}{3} - 3 x^{2} + 9 x - \frac{26}{3}$

Step 4

Verify if $f^{- 1} (x) = \frac{x^{3}}{3} - 3 x^{2} + 9 x - \frac{26}{3}$ is the inverse of $f (x) = \sqrt[3]{3 x - 1} + 3$ .

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Step 4.1

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

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

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

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{{(\sqrt[3]{3 x - 1} + 3)}^{3}}{3} - 3 {(\sqrt[3]{3 x - 1} + 3)}^{2} + 9 (\sqrt[3]{3 x - 1} + 3) - \frac{26}{3}$

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

Rewrite ${(\sqrt[3]{3 x - 1} + 3)}^{2}$ as $(\sqrt[3]{3 x - 1} + 3) (\sqrt[3]{3 x - 1} + 3)$ .

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{{(\sqrt[3]{3 x - 1} + 3)}^{3}}{3} - 3 ((\sqrt[3]{3 x - 1} + 3) (\sqrt[3]{3 x - 1} + 3)) + 9 (\sqrt[3]{3 x - 1} + 3) - \frac{26}{3}$

Step 4.2.3.2

Expand $(\sqrt[3]{3 x - 1} + 3) (\sqrt[3]{3 x - 1} + 3)$ using the FOIL Method.

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Step 4.2.3.2.1

Apply the distributive property.

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{{(\sqrt[3]{3 x - 1} + 3)}^{3}}{3} - 3 (\sqrt[3]{3 x - 1} (\sqrt[3]{3 x - 1} + 3) + 3 (\sqrt[3]{3 x - 1} + 3)) + 9 (\sqrt[3]{3 x - 1} + 3) - \frac{26}{3}$

Step 4.2.3.2.2

Apply the distributive property.

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{{(\sqrt[3]{3 x - 1} + 3)}^{3}}{3} - 3 (\sqrt[3]{3 x - 1} \sqrt[3]{3 x - 1} + \sqrt[3]{3 x - 1} \cdot 3 + 3 (\sqrt[3]{3 x - 1} + 3)) + 9 (\sqrt[3]{3 x - 1} + 3) - \frac{26}{3}$

Step 4.2.3.2.3

Apply the distributive property.

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{{(\sqrt[3]{3 x - 1} + 3)}^{3}}{3} - 3 (\sqrt[3]{3 x - 1} \sqrt[3]{3 x - 1} + \sqrt[3]{3 x - 1} \cdot 3 + 3 \sqrt[3]{3 x - 1} + 3 \cdot 3) + 9 (\sqrt[3]{3 x - 1} + 3) - \frac{26}{3}$

Step 4.2.3.3

Simplify and combine like terms.

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Step 4.2.3.3.1

Simplify each term.

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Step 4.2.3.3.1.1

Multiply $\sqrt[3]{3 x - 1} \sqrt[3]{3 x - 1}$ .

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Step 4.2.3.3.1.1.1

Raise $\sqrt[3]{3 x - 1}$ to the power of $1$ .

Step 4.2.3.3.1.1.2

Raise $\sqrt[3]{3 x - 1}$ to the power of $1$ .

Step 4.2.3.3.1.1.3

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

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{{(\sqrt[3]{3 x - 1} + 3)}^{3}}{3} - 3 ({\sqrt[3]{3 x - 1}}^{1 + 1} + \sqrt[3]{3 x - 1} \cdot 3 + 3 \sqrt[3]{3 x - 1} + 3 \cdot 3) + 9 (\sqrt[3]{3 x - 1} + 3) - \frac{26}{3}$

Step 4.2.3.3.1.1.4

Add $1$ and $1$ .

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{{(\sqrt[3]{3 x - 1} + 3)}^{3}}{3} - 3 ({\sqrt[3]{3 x - 1}}^{2} + \sqrt[3]{3 x - 1} \cdot 3 + 3 \sqrt[3]{3 x - 1} + 3 \cdot 3) + 9 (\sqrt[3]{3 x - 1} + 3) - \frac{26}{3}$

Step 4.2.3.3.1.2

Rewrite ${\sqrt[3]{3 x - 1}}^{2}$ as $\sqrt[3]{{(3 x - 1)}^{2}}$ .

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{{(\sqrt[3]{3 x - 1} + 3)}^{3}}{3} - 3 (\sqrt[3]{{(3 x - 1)}^{2}} + \sqrt[3]{3 x - 1} \cdot 3 + 3 \sqrt[3]{3 x - 1} + 3 \cdot 3) + 9 (\sqrt[3]{3 x - 1} + 3) - \frac{26}{3}$

Step 4.2.3.3.1.3

Move $3$ to the left of $\sqrt[3]{3 x - 1}$ .

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{{(\sqrt[3]{3 x - 1} + 3)}^{3}}{3} - 3 (\sqrt[3]{{(3 x - 1)}^{2}} + 3 \cdot \sqrt[3]{3 x - 1} + 3 \sqrt[3]{3 x - 1} + 3 \cdot 3) + 9 (\sqrt[3]{3 x - 1} + 3) - \frac{26}{3}$

Step 4.2.3.3.1.4

Multiply $3$ by $3$ .

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{{(\sqrt[3]{3 x - 1} + 3)}^{3}}{3} - 3 (\sqrt[3]{{(3 x - 1)}^{2}} + 3 \sqrt[3]{3 x - 1} + 3 \sqrt[3]{3 x - 1} + 9) + 9 (\sqrt[3]{3 x - 1} + 3) - \frac{26}{3}$

Step 4.2.3.3.2

Add $3 \sqrt[3]{3 x - 1}$ and $3 \sqrt[3]{3 x - 1}$ .

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{{(\sqrt[3]{3 x - 1} + 3)}^{3}}{3} - 3 (\sqrt[3]{{(3 x - 1)}^{2}} + 6 \sqrt[3]{3 x - 1} + 9) + 9 (\sqrt[3]{3 x - 1} + 3) - \frac{26}{3}$

Step 4.2.3.4

Apply the distributive property.

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{{(\sqrt[3]{3 x - 1} + 3)}^{3}}{3} - 3 \sqrt[3]{{(3 x - 1)}^{2}} - 3 (6 \sqrt[3]{3 x - 1}) - 3 \cdot 9 + 9 (\sqrt[3]{3 x - 1} + 3) - \frac{26}{3}$

Step 4.2.3.5

Simplify.

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Step 4.2.3.5.1

Multiply $6$ by $- 3$ .

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{{(\sqrt[3]{3 x - 1} + 3)}^{3}}{3} - 3 \sqrt[3]{{(3 x - 1)}^{2}} - 18 \sqrt[3]{3 x - 1} - 3 \cdot 9 + 9 (\sqrt[3]{3 x - 1} + 3) - \frac{26}{3}$

Step 4.2.3.5.2

Multiply $- 3$ by $9$ .

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{{(\sqrt[3]{3 x - 1} + 3)}^{3}}{3} - 3 \sqrt[3]{{(3 x - 1)}^{2}} - 18 \sqrt[3]{3 x - 1} - 27 + 9 (\sqrt[3]{3 x - 1} + 3) - \frac{26}{3}$

Step 4.2.3.6

Apply the distributive property.

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{{(\sqrt[3]{3 x - 1} + 3)}^{3}}{3} - 3 \sqrt[3]{{(3 x - 1)}^{2}} - 18 \sqrt[3]{3 x - 1} - 27 + 9 \sqrt[3]{3 x - 1} + 9 \cdot 3 - \frac{26}{3}$

Step 4.2.3.7

Multiply $9$ by $3$ .

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{{(\sqrt[3]{3 x - 1} + 3)}^{3}}{3} - 3 \sqrt[3]{{(3 x - 1)}^{2}} - 18 \sqrt[3]{3 x - 1} - 27 + 9 \sqrt[3]{3 x - 1} + 27 - \frac{26}{3}$

Step 4.2.4

Combine the opposite terms in $\frac{{(\sqrt[3]{3 x - 1} + 3)}^{3}}{3} - 3 \sqrt[3]{{(3 x - 1)}^{2}} - 18 \sqrt[3]{3 x - 1} - 27 + 9 \sqrt[3]{3 x - 1} + 27 - \frac{26}{3}$ .

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Step 4.2.4.1

Add $- 27$ and $27$ .

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{{(\sqrt[3]{3 x - 1} + 3)}^{3}}{3} - 3 \sqrt[3]{{(3 x - 1)}^{2}} - 18 \sqrt[3]{3 x - 1} + 0 + 9 \sqrt[3]{3 x - 1} - \frac{26}{3}$

Step 4.2.4.2

Add $\frac{{(\sqrt[3]{3 x - 1} + 3)}^{3}}{3} - 3 \sqrt[3]{{(3 x - 1)}^{2}} - 18 \sqrt[3]{3 x - 1}$ and $0$ .

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{{(\sqrt[3]{3 x - 1} + 3)}^{3}}{3} - 3 \sqrt[3]{{(3 x - 1)}^{2}} - 18 \sqrt[3]{3 x - 1} + 9 \sqrt[3]{3 x - 1} - \frac{26}{3}$

Step 4.2.5

To write $- 3 \sqrt[3]{{(3 x - 1)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{{(\sqrt[3]{3 x - 1} + 3)}^{3}}{3} - 3 \sqrt[3]{{(3 x - 1)}^{2}} \cdot \frac{3}{3} - 18 \sqrt[3]{3 x - 1} + 9 \sqrt[3]{3 x - 1} - \frac{26}{3}$

Step 4.2.6

Simplify terms.

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Step 4.2.6.1

Combine $- 3 \sqrt[3]{{(3 x - 1)}^{2}}$ and $\frac{3}{3}$ .

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{{(\sqrt[3]{3 x - 1} + 3)}^{3}}{3} + \frac{- 3 \sqrt[3]{{(3 x - 1)}^{2}} \cdot 3}{3} - 18 \sqrt[3]{3 x - 1} + 9 \sqrt[3]{3 x - 1} - \frac{26}{3}$

Step 4.2.6.2

Combine the numerators over the common denominator.

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{{(\sqrt[3]{3 x - 1} + 3)}^{3} - 3 \sqrt[3]{{(3 x - 1)}^{2}} \cdot 3}{3} - 18 \sqrt[3]{3 x - 1} + 9 \sqrt[3]{3 x - 1} - \frac{26}{3}$

Step 4.2.7

Simplify the numerator.

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Step 4.2.7.1

Use the Binomial Theorem.

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{{\sqrt[3]{3 x - 1}}^{3} + 3 {\sqrt[3]{3 x - 1}}^{2} \cdot 3 + 3 \sqrt[3]{3 x - 1} \cdot 3^{2} + 3^{3} - 3 \sqrt[3]{{(3 x - 1)}^{2}} \cdot 3}{3} - 18 \sqrt[3]{3 x - 1} + 9 \sqrt[3]{3 x - 1} - \frac{26}{3}$

Step 4.2.7.2

Simplify each term.

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Step 4.2.7.2.1

Rewrite ${\sqrt[3]{3 x - 1}}^{3}$ as $3 x - 1$ .

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Step 4.2.7.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{3 x - 1}$ as ${(3 x - 1)}^{\frac{1}{3}}$ .

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{{({(3 x - 1)}^{\frac{1}{3}})}^{3} + 3 {\sqrt[3]{3 x - 1}}^{2} \cdot 3 + 3 \sqrt[3]{3 x - 1} \cdot 3^{2} + 3^{3} - 3 \sqrt[3]{{(3 x - 1)}^{2}} \cdot 3}{3} - 18 \sqrt[3]{3 x - 1} + 9 \sqrt[3]{3 x - 1} - \frac{26}{3}$

Step 4.2.7.2.1.2

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

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{{(3 x - 1)}^{\frac{1}{3} \cdot 3} + 3 {\sqrt[3]{3 x - 1}}^{2} \cdot 3 + 3 \sqrt[3]{3 x - 1} \cdot 3^{2} + 3^{3} - 3 \sqrt[3]{{(3 x - 1)}^{2}} \cdot 3}{3} - 18 \sqrt[3]{3 x - 1} + 9 \sqrt[3]{3 x - 1} - \frac{26}{3}$

Step 4.2.7.2.1.3

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

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{{(3 x - 1)}^{\frac{3}{3}} + 3 {\sqrt[3]{3 x - 1}}^{2} \cdot 3 + 3 \sqrt[3]{3 x - 1} \cdot 3^{2} + 3^{3} - 3 \sqrt[3]{{(3 x - 1)}^{2}} \cdot 3}{3} - 18 \sqrt[3]{3 x - 1} + 9 \sqrt[3]{3 x - 1} - \frac{26}{3}$

Step 4.2.7.2.1.4

Cancel the common factor of $3$ .

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Step 4.2.7.2.1.4.1

Cancel the common factor.

Step 4.2.7.2.1.4.2

Rewrite the expression.

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{(3 x - 1) + 3 {\sqrt[3]{3 x - 1}}^{2} \cdot 3 + 3 \sqrt[3]{3 x - 1} \cdot 3^{2} + 3^{3} - 3 \sqrt[3]{{(3 x - 1)}^{2}} \cdot 3}{3} - 18 \sqrt[3]{3 x - 1} + 9 \sqrt[3]{3 x - 1} - \frac{26}{3}$

Step 4.2.7.2.1.5

Simplify.

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{3 x - 1 + 3 {\sqrt[3]{3 x - 1}}^{2} \cdot 3 + 3 \sqrt[3]{3 x - 1} \cdot 3^{2} + 3^{3} - 3 \sqrt[3]{{(3 x - 1)}^{2}} \cdot 3}{3} - 18 \sqrt[3]{3 x - 1} + 9 \sqrt[3]{3 x - 1} - \frac{26}{3}$

Step 4.2.7.2.2

Rewrite ${\sqrt[3]{3 x - 1}}^{2}$ as $\sqrt[3]{{(3 x - 1)}^{2}}$ .

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{3 x - 1 + 3 \sqrt[3]{{(3 x - 1)}^{2}} \cdot 3 + 3 \sqrt[3]{3 x - 1} \cdot 3^{2} + 3^{3} - 3 \sqrt[3]{{(3 x - 1)}^{2}} \cdot 3}{3} - 18 \sqrt[3]{3 x - 1} + 9 \sqrt[3]{3 x - 1} - \frac{26}{3}$

Step 4.2.7.2.3

Multiply $3$ by $3$ .

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{3 x - 1 + 9 \sqrt[3]{{(3 x - 1)}^{2}} + 3 \sqrt[3]{3 x - 1} \cdot 3^{2} + 3^{3} - 3 \sqrt[3]{{(3 x - 1)}^{2}} \cdot 3}{3} - 18 \sqrt[3]{3 x - 1} + 9 \sqrt[3]{3 x - 1} - \frac{26}{3}$

Step 4.2.7.2.4

Multiply $3$ by $3^{2}$ by adding the exponents.

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Step 4.2.7.2.4.1

Move $3^{2}$ .

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{3 x - 1 + 9 \sqrt[3]{{(3 x - 1)}^{2}} + 3^{2} \cdot (3 \sqrt[3]{3 x - 1}) + 3^{3} - 3 \sqrt[3]{{(3 x - 1)}^{2}} \cdot 3}{3} - 18 \sqrt[3]{3 x - 1} + 9 \sqrt[3]{3 x - 1} - \frac{26}{3}$

Step 4.2.7.2.4.2

Multiply $3^{2}$ by $3$ .

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Step 4.2.7.2.4.2.1

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

Step 4.2.7.2.4.2.2

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

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{3 x - 1 + 9 \sqrt[3]{{(3 x - 1)}^{2}} + 3^{2 + 1} \sqrt[3]{3 x - 1} + 3^{3} - 3 \sqrt[3]{{(3 x - 1)}^{2}} \cdot 3}{3} - 18 \sqrt[3]{3 x - 1} + 9 \sqrt[3]{3 x - 1} - \frac{26}{3}$

Step 4.2.7.2.4.3

Add $2$ and $1$ .

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{3 x - 1 + 9 \sqrt[3]{{(3 x - 1)}^{2}} + 3^{3} \sqrt[3]{3 x - 1} + 3^{3} - 3 \sqrt[3]{{(3 x - 1)}^{2}} \cdot 3}{3} - 18 \sqrt[3]{3 x - 1} + 9 \sqrt[3]{3 x - 1} - \frac{26}{3}$

Step 4.2.7.2.5

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

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{3 x - 1 + 9 \sqrt[3]{{(3 x - 1)}^{2}} + 27 \sqrt[3]{3 x - 1} + 3^{3} - 3 \sqrt[3]{{(3 x - 1)}^{2}} \cdot 3}{3} - 18 \sqrt[3]{3 x - 1} + 9 \sqrt[3]{3 x - 1} - \frac{26}{3}$

Step 4.2.7.2.6

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

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{3 x - 1 + 9 \sqrt[3]{{(3 x - 1)}^{2}} + 27 \sqrt[3]{3 x - 1} + 27 - 3 \sqrt[3]{{(3 x - 1)}^{2}} \cdot 3}{3} - 18 \sqrt[3]{3 x - 1} + 9 \sqrt[3]{3 x - 1} - \frac{26}{3}$

Step 4.2.7.3

Add $- 1$ and $27$ .

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{3 x + 26 + 9 \sqrt[3]{{(3 x - 1)}^{2}} + 27 \sqrt[3]{3 x - 1} - 3 \sqrt[3]{{(3 x - 1)}^{2}} \cdot 3}{3} - 18 \sqrt[3]{3 x - 1} + 9 \sqrt[3]{3 x - 1} - \frac{26}{3}$

Step 4.2.7.4

Multiply $3$ by $- 3$ .

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{3 x + 26 + 9 \sqrt[3]{{(3 x - 1)}^{2}} + 27 \sqrt[3]{3 x - 1} - 9 \sqrt[3]{{(3 x - 1)}^{2}}}{3} - 18 \sqrt[3]{3 x - 1} + 9 \sqrt[3]{3 x - 1} - \frac{26}{3}$

Step 4.2.7.5

Subtract $9 \sqrt[3]{{(3 x - 1)}^{2}}$ from $9 \sqrt[3]{{(3 x - 1)}^{2}}$ .

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{3 x + 26 + 0 + 27 \sqrt[3]{3 x - 1}}{3} - 18 \sqrt[3]{3 x - 1} + 9 \sqrt[3]{3 x - 1} - \frac{26}{3}$

Step 4.2.7.6

Add $3 x$ and $0$ .

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = \frac{3 x + 26 + 27 \sqrt[3]{3 x - 1}}{3} - 18 \sqrt[3]{3 x - 1} + 9 \sqrt[3]{3 x - 1} - \frac{26}{3}$

Step 4.2.8

Simplify terms.

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Step 4.2.8.1

Combine the numerators over the common denominator.

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = - 18 \sqrt[3]{3 x - 1} + 9 \sqrt[3]{3 x - 1} + \frac{3 x + 26 + 27 \sqrt[3]{3 x - 1} - 26}{3}$

Step 4.2.8.2

Combine the opposite terms in $3 x + 26 + 27 \sqrt[3]{3 x - 1} - 26$ .

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Step 4.2.8.2.1

Subtract $26$ from $26$ .

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = - 18 \sqrt[3]{3 x - 1} + 9 \sqrt[3]{3 x - 1} + \frac{3 x + 0 + 27 \sqrt[3]{3 x - 1}}{3}$

Step 4.2.8.2.2

Add $3 x$ and $0$ .

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = - 18 \sqrt[3]{3 x - 1} + 9 \sqrt[3]{3 x - 1} + \frac{3 x + 27 \sqrt[3]{3 x - 1}}{3}$

Step 4.2.8.3

Cancel the common factor of $3 x + 27 \sqrt[3]{3 x - 1}$ and $3$ .

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Step 4.2.8.3.1

Factor $3$ out of $3 x$ .

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = - 18 \sqrt[3]{3 x - 1} + 9 \sqrt[3]{3 x - 1} + \frac{3 (x) + 27 \sqrt[3]{3 x - 1}}{3}$

Step 4.2.8.3.2

Factor $3$ out of $27 \sqrt[3]{3 x - 1}$ .

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = - 18 \sqrt[3]{3 x - 1} + 9 \sqrt[3]{3 x - 1} + \frac{3 (x) + 3 (9 \sqrt[3]{3 x - 1})}{3}$

Step 4.2.8.3.3

Factor $3$ out of $3 (x) + 3 (9 \sqrt[3]{3 x - 1})$ .

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = - 18 \sqrt[3]{3 x - 1} + 9 \sqrt[3]{3 x - 1} + \frac{3 (x + 9 \sqrt[3]{3 x - 1})}{3}$

Step 4.2.8.3.4

Cancel the common factors.

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Step 4.2.8.3.4.1

Factor $3$ out of $3$ .

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = - 18 \sqrt[3]{3 x - 1} + 9 \sqrt[3]{3 x - 1} + \frac{3 (x + 9 \sqrt[3]{3 x - 1})}{3 (1)}$

Step 4.2.8.3.4.2

Cancel the common factor.

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = - 18 \sqrt[3]{3 x - 1} + 9 \sqrt[3]{3 x - 1} + \frac{3 (x + 9 \sqrt[3]{3 x - 1})}{3 \cdot 1}$

Step 4.2.8.3.4.3

Rewrite the expression.

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = - 18 \sqrt[3]{3 x - 1} + 9 \sqrt[3]{3 x - 1} + \frac{x + 9 \sqrt[3]{3 x - 1}}{1}$

Step 4.2.8.3.4.4

Divide $x + 9 \sqrt[3]{3 x - 1}$ by $1$ .

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = - 18 \sqrt[3]{3 x - 1} + 9 \sqrt[3]{3 x - 1} + x + 9 \sqrt[3]{3 x - 1}$

Step 4.2.8.4

Add $- 18 \sqrt[3]{3 x - 1}$ and $9 \sqrt[3]{3 x - 1}$ .

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = - 9 \sqrt[3]{3 x - 1} + x + 9 \sqrt[3]{3 x - 1}$

Step 4.2.8.5

Combine the opposite terms in $- 9 \sqrt[3]{3 x - 1} + x + 9 \sqrt[3]{3 x - 1}$ .

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Step 4.2.8.5.1

Add $- 9 \sqrt[3]{3 x - 1}$ and $9 \sqrt[3]{3 x - 1}$ .

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = x + 0$

Step 4.2.8.5.2

Add $x$ and $0$ .

$f^{- 1} (\sqrt[3]{3 x - 1} + 3) = x$

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

Evaluate $f (\frac{x^{3}}{3} - 3 x^{2} + 9 x - \frac{26}{3})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x^{3}}{3} - 3 x^{2} + 9 x - \frac{26}{3}) = \sqrt[3]{3 (\frac{x^{3}}{3} - 3 x^{2} + 9 x - \frac{26}{3}) - 1} + 3$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Apply the distributive property.

$f (\frac{x^{3}}{3} - 3 x^{2} + 9 x - \frac{26}{3}) = \sqrt[3]{3 (\frac{x^{3}}{3}) + 3 (- 3 x^{2}) + 3 (9 x) + 3 (- \frac{26}{3}) - 1} + 3$

Step 4.3.3.2

Simplify.

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Step 4.3.3.2.1

Cancel the common factor of $3$ .

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Step 4.3.3.2.1.1

Cancel the common factor.

$f (\frac{x^{3}}{3} - 3 x^{2} + 9 x - \frac{26}{3}) = \sqrt[3]{3 (\frac{x^{3}}{3}) + 3 (- 3 x^{2}) + 3 (9 x) + 3 (- \frac{26}{3}) - 1} + 3$

Step 4.3.3.2.1.2

Rewrite the expression.

$f (\frac{x^{3}}{3} - 3 x^{2} + 9 x - \frac{26}{3}) = \sqrt[3]{x^{3} + 3 (- 3 x^{2}) + 3 (9 x) + 3 (- \frac{26}{3}) - 1} + 3$

Step 4.3.3.2.2

Multiply $- 3$ by $3$ .

$f (\frac{x^{3}}{3} - 3 x^{2} + 9 x - \frac{26}{3}) = \sqrt[3]{x^{3} - 9 x^{2} + 3 (9 x) + 3 (- \frac{26}{3}) - 1} + 3$

Step 4.3.3.2.3

Multiply $9$ by $3$ .

$f (\frac{x^{3}}{3} - 3 x^{2} + 9 x - \frac{26}{3}) = \sqrt[3]{x^{3} - 9 x^{2} + 27 x + 3 (- \frac{26}{3}) - 1} + 3$

Step 4.3.3.2.4

Cancel the common factor of $3$ .

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Step 4.3.3.2.4.1

Move the leading negative in $- \frac{26}{3}$ into the numerator.

$f (\frac{x^{3}}{3} - 3 x^{2} + 9 x - \frac{26}{3}) = \sqrt[3]{x^{3} - 9 x^{2} + 27 x + 3 (\frac{- 26}{3}) - 1} + 3$

Step 4.3.3.2.4.2

Cancel the common factor.

$f (\frac{x^{3}}{3} - 3 x^{2} + 9 x - \frac{26}{3}) = \sqrt[3]{x^{3} - 9 x^{2} + 27 x + 3 (\frac{- 26}{3}) - 1} + 3$

Step 4.3.3.2.4.3

Rewrite the expression.

$f (\frac{x^{3}}{3} - 3 x^{2} + 9 x - \frac{26}{3}) = \sqrt[3]{x^{3} - 9 x^{2} + 27 x - 26 - 1} + 3$

Step 4.3.3.3

Subtract $1$ from $- 26$ .

$f (\frac{x^{3}}{3} - 3 x^{2} + 9 x - \frac{26}{3}) = \sqrt[3]{x^{3} - 9 x^{2} + 27 x - 27} + 3$

Step 4.3.3.4

Rewrite $x^{3} - 9 x^{2} + 27 x - 27$ in a factored form.

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Step 4.3.3.4.1

Factor $x^{3} - 9 x^{2} + 27 x - 27$ using the rational roots test.

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Step 4.3.3.4.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 27, \pm 3, \pm 9$

$q = \pm 1$

Step 4.3.3.4.1.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 27, \pm 3, \pm 9$

Step 4.3.3.4.1.3

Substitute $3$ and simplify the expression. In this case, the expression is equal to $0$ so $3$ is a root of the polynomial.

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Step 4.3.3.4.1.3.1

Substitute $3$ into the polynomial.

$3^{3} - 9 \cdot 3^{2} + 27 \cdot 3 - 27$

Step 4.3.3.4.1.3.2

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

$27 - 9 \cdot 3^{2} + 27 \cdot 3 - 27$

Step 4.3.3.4.1.3.3

Raise $3$ to the power of $2$ .

$27 - 9 \cdot 9 + 27 \cdot 3 - 27$

Step 4.3.3.4.1.3.4

Multiply $- 9$ by $9$ .

$27 - 81 + 27 \cdot 3 - 27$

Step 4.3.3.4.1.3.5

Subtract $81$ from $27$ .

$- 54 + 27 \cdot 3 - 27$

Step 4.3.3.4.1.3.6

Multiply $27$ by $3$ .

$- 54 + 81 - 27$

Step 4.3.3.4.1.3.7

Add $- 54$ and $81$ .

$27 - 27$

Step 4.3.3.4.1.3.8

Subtract $27$ from $27$ .

$0$

Step 4.3.3.4.1.4

Since $3$ is a known root, divide the polynomial by $x - 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 9 x^{2} + 27 x - 27}{x - 3}$

Step 4.3.3.4.1.5

Divide $x^{3} - 9 x^{2} + 27 x - 27$ by $x - 3$ .

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Step 4.3.3.4.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$3$

$x^{3}$

$9 x^{2}$

$27 x$

$27$

Step 4.3.3.4.1.5.2

Divide the highest order term in the dividend $x^{3}$ by the highest order term in divisor $x$ .

					$x^{2}$
	$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$27 x$	-	$27$

Step 4.3.3.4.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$27 x$	-	$27$
			+	$x^{3}$	-	$3 x^{2}$

Step 4.3.3.4.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - 3 x^{2}$

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$27 x$	-	$27$
			-	$x^{3}$	+	$3 x^{2}$

Step 4.3.3.4.1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$27 x$	-	$27$
			-	$x^{3}$	+	$3 x^{2}$
					-	$6 x^{2}$

Step 4.3.3.4.1.5.6

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$27 x$	-	$27$
			-	$x^{3}$	+	$3 x^{2}$
					-	$6 x^{2}$	+	$27 x$

Step 4.3.3.4.1.5.7

Divide the highest order term in the dividend $- 6 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$6 x$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$27 x$	-	$27$
			-	$x^{3}$	+	$3 x^{2}$
					-	$6 x^{2}$	+	$27 x$

Step 4.3.3.4.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$6 x$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$27 x$	-	$27$
			-	$x^{3}$	+	$3 x^{2}$
					-	$6 x^{2}$	+	$27 x$
					-	$6 x^{2}$	+	$18 x$

Step 4.3.3.4.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x^{2} + 18 x$

				$x^{2}$	-	$6 x$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$27 x$	-	$27$
			-	$x^{3}$	+	$3 x^{2}$
					-	$6 x^{2}$	+	$27 x$
					+	$6 x^{2}$	-	$18 x$

Step 4.3.3.4.1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	-	$6 x$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$27 x$	-	$27$
			-	$x^{3}$	+	$3 x^{2}$
					-	$6 x^{2}$	+	$27 x$
					+	$6 x^{2}$	-	$18 x$
							+	$9 x$

Step 4.3.3.4.1.5.11

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$	-	$6 x$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$27 x$	-	$27$
			-	$x^{3}$	+	$3 x^{2}$
					-	$6 x^{2}$	+	$27 x$
					+	$6 x^{2}$	-	$18 x$
							+	$9 x$	-	$27$

Step 4.3.3.4.1.5.12

Divide the highest order term in the dividend $9 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$6 x$	+	$9$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$27 x$	-	$27$
			-	$x^{3}$	+	$3 x^{2}$
					-	$6 x^{2}$	+	$27 x$
					+	$6 x^{2}$	-	$18 x$
							+	$9 x$	-	$27$

Step 4.3.3.4.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$6 x$	+	$9$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$27 x$	-	$27$
			-	$x^{3}$	+	$3 x^{2}$
					-	$6 x^{2}$	+	$27 x$
					+	$6 x^{2}$	-	$18 x$
							+	$9 x$	-	$27$
							+	$9 x$	-	$27$

Step 4.3.3.4.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $9 x - 27$

				$x^{2}$	-	$6 x$	+	$9$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$27 x$	-	$27$
			-	$x^{3}$	+	$3 x^{2}$
					-	$6 x^{2}$	+	$27 x$
					+	$6 x^{2}$	-	$18 x$
							+	$9 x$	-	$27$
							-	$9 x$	+	$27$

Step 4.3.3.4.1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	-	$6 x$	+	$9$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$27 x$	-	$27$
			-	$x^{3}$	+	$3 x^{2}$
					-	$6 x^{2}$	+	$27 x$
					+	$6 x^{2}$	-	$18 x$
							+	$9 x$	-	$27$
							-	$9 x$	+	$27$
										$0$

Step 4.3.3.4.1.5.16

Since the remander is $0$ , the final answer is the quotient.

$x^{2} - 6 x + 9$

Step 4.3.3.4.1.6

Write $x^{3} - 9 x^{2} + 27 x - 27$ as a set of factors.

$f (\frac{x^{3}}{3} - 3 x^{2} + 9 x - \frac{26}{3}) = \sqrt[3]{(x - 3) (x^{2} - 6 x + 9)} + 3$

Step 4.3.3.4.2

Factor using the perfect square rule.

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Step 4.3.3.4.2.1

Rewrite $9$ as $3^{2}$ .

$f (\frac{x^{3}}{3} - 3 x^{2} + 9 x - \frac{26}{3}) = \sqrt[3]{(x - 3) (x^{2} - 6 x + 3^{2})} + 3$

Step 4.3.3.4.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 x = 2 \cdot x \cdot 3$

Step 4.3.3.4.2.3

Rewrite the polynomial.

$f (\frac{x^{3}}{3} - 3 x^{2} + 9 x - \frac{26}{3}) = \sqrt[3]{(x - 3) (x^{2} - 2 \cdot x \cdot 3 + 3^{2})} + 3$

Step 4.3.3.4.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 3$ .

$f (\frac{x^{3}}{3} - 3 x^{2} + 9 x - \frac{26}{3}) = \sqrt[3]{(x - 3) {(x - 3)}^{2}} + 3$

Step 4.3.3.4.3

Combine like factors.

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Step 4.3.3.4.3.1

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

$f (\frac{x^{3}}{3} - 3 x^{2} + 9 x - \frac{26}{3}) = \sqrt[3]{(x - 3) {(x - 3)}^{2}} + 3$

Step 4.3.3.4.3.2

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

$f (\frac{x^{3}}{3} - 3 x^{2} + 9 x - \frac{26}{3}) = \sqrt[3]{{(x - 3)}^{1 + 2}} + 3$

Step 4.3.3.4.3.3

Add $1$ and $2$ .

$f (\frac{x^{3}}{3} - 3 x^{2} + 9 x - \frac{26}{3}) = \sqrt[3]{{(x - 3)}^{3}} + 3$

Step 4.3.3.5

Pull terms out from under the radical, assuming real numbers.

$f (\frac{x^{3}}{3} - 3 x^{2} + 9 x - \frac{26}{3}) = x - 3 + 3$

Step 4.3.4

Combine the opposite terms in $x - 3 + 3$ .

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Step 4.3.4.1

Add $- 3$ and $3$ .

$f (\frac{x^{3}}{3} - 3 x^{2} + 9 x - \frac{26}{3}) = x + 0$

Step 4.3.4.2

Add $x$ and $0$ .

$f (\frac{x^{3}}{3} - 3 x^{2} + 9 x - \frac{26}{3}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x^{3}}{3} - 3 x^{2} + 9 x - \frac{26}{3}$ is the inverse of $f (x) = \sqrt[3]{3 x - 1} + 3$ .

$f^{- 1} (x) = \frac{x^{3}}{3} - 3 x^{2} + 9 x - \frac{26}{3}$