Algebra Examples

$\sqrt[3]{x} - 6$

Step 1

Interchange the variables.

$x = \sqrt[3]{y} - 6$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\sqrt[3]{y} - 6 = x$ .

$\sqrt[3]{y} - 6 = x$

Step 2.2

Add $6$ to both sides of the equation.

$\sqrt[3]{y} = x + 6$

Step 2.3

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

${\sqrt[3]{y}}^{3} = {(x + 6)}^{3}$

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 $\sqrt[3]{y}$ as $y^{\frac{1}{3}}$ .

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

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

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

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Step 2.4.2.1.1

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

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Step 2.4.2.1.1.1

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

$y^{\frac{1}{3} \cdot 3} = {(x + 6)}^{3}$

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.

$y^{\frac{1}{3} \cdot 3} = {(x + 6)}^{3}$

Step 2.4.2.1.1.2.2

Rewrite the expression.

$y^{1} = {(x + 6)}^{3}$

Step 2.4.2.1.2

Simplify.

$y = {(x + 6)}^{3}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Simplify ${(x + 6)}^{3}$ .

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Step 2.4.3.1.1

Use the Binomial Theorem.

$y = x^{3} + 3 x^{2} \cdot 6 + 3 x \cdot 6^{2} + 6^{3}$

Step 2.4.3.1.2

Simplify each term.

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Step 2.4.3.1.2.1

Multiply $6$ by $3$ .

$y = x^{3} + 18 x^{2} + 3 x \cdot 6^{2} + 6^{3}$

Step 2.4.3.1.2.2

Raise $6$ to the power of $2$ .

$y = x^{3} + 18 x^{2} + 3 x \cdot 36 + 6^{3}$

Step 2.4.3.1.2.3

Multiply $36$ by $3$ .

$y = x^{3} + 18 x^{2} + 108 x + 6^{3}$

Step 2.4.3.1.2.4

Raise $6$ to the power of $3$ .

$y = x^{3} + 18 x^{2} + 108 x + 216$

Step 3

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

$f^{- 1} (x) = x^{3} + 18 x^{2} + 108 x + 216$

Step 4

Verify if $f^{- 1} (x) = x^{3} + 18 x^{2} + 108 x + 216$ is the inverse of $f (x) = \sqrt[3]{x} - 6$ .

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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 $f^{- 1} (\sqrt[3]{x} - 6)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\sqrt[3]{x} - 6) = {(\sqrt[3]{x} - 6)}^{3} + 18 {(\sqrt[3]{x} - 6)}^{2} + 108 (\sqrt[3]{x} - 6) + 216$

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

Use the Binomial Theorem.

$f^{- 1} (\sqrt[3]{x} - 6) = {\sqrt[3]{x}}^{3} + 3 {\sqrt[3]{x}}^{2} \cdot - 6 + 3 \sqrt[3]{x} {(- 6)}^{2} + {(- 6)}^{3} + 18 {(\sqrt[3]{x} - 6)}^{2} + 108 (\sqrt[3]{x} - 6) + 216$

Step 4.2.3.2

Simplify each term.

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Step 4.2.3.2.1

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

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Step 4.2.3.2.1.1

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

$f^{- 1} (\sqrt[3]{x} - 6) = {(x^{\frac{1}{3}})}^{3} + 3 {\sqrt[3]{x}}^{2} \cdot - 6 + 3 \sqrt[3]{x} {(- 6)}^{2} + {(- 6)}^{3} + 18 {(\sqrt[3]{x} - 6)}^{2} + 108 (\sqrt[3]{x} - 6) + 216$

Step 4.2.3.2.1.2

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

$f^{- 1} (\sqrt[3]{x} - 6) = x^{\frac{1}{3} \cdot 3} + 3 {\sqrt[3]{x}}^{2} \cdot - 6 + 3 \sqrt[3]{x} {(- 6)}^{2} + {(- 6)}^{3} + 18 {(\sqrt[3]{x} - 6)}^{2} + 108 (\sqrt[3]{x} - 6) + 216$

Step 4.2.3.2.1.3

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

$f^{- 1} (\sqrt[3]{x} - 6) = x^{\frac{3}{3}} + 3 {\sqrt[3]{x}}^{2} \cdot - 6 + 3 \sqrt[3]{x} {(- 6)}^{2} + {(- 6)}^{3} + 18 {(\sqrt[3]{x} - 6)}^{2} + 108 (\sqrt[3]{x} - 6) + 216$

Step 4.2.3.2.1.4

Cancel the common factor of $3$ .

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Step 4.2.3.2.1.4.1

Cancel the common factor.

$f^{- 1} (\sqrt[3]{x} - 6) = x^{\frac{3}{3}} + 3 {\sqrt[3]{x}}^{2} \cdot - 6 + 3 \sqrt[3]{x} {(- 6)}^{2} + {(- 6)}^{3} + 18 {(\sqrt[3]{x} - 6)}^{2} + 108 (\sqrt[3]{x} - 6) + 216$

Step 4.2.3.2.1.4.2

Rewrite the expression.

$f^{- 1} (\sqrt[3]{x} - 6) = x + 3 {\sqrt[3]{x}}^{2} \cdot - 6 + 3 \sqrt[3]{x} {(- 6)}^{2} + {(- 6)}^{3} + 18 {(\sqrt[3]{x} - 6)}^{2} + 108 (\sqrt[3]{x} - 6) + 216$

Step 4.2.3.2.1.5

Simplify.

$f^{- 1} (\sqrt[3]{x} - 6) = x + 3 {\sqrt[3]{x}}^{2} \cdot - 6 + 3 \sqrt[3]{x} {(- 6)}^{2} + {(- 6)}^{3} + 18 {(\sqrt[3]{x} - 6)}^{2} + 108 (\sqrt[3]{x} - 6) + 216$

Step 4.2.3.2.2

Rewrite ${\sqrt[3]{x}}^{2}$ as $\sqrt[3]{x^{2}}$ .

$f^{- 1} (\sqrt[3]{x} - 6) = x + 3 \sqrt[3]{x^{2}} \cdot - 6 + 3 \sqrt[3]{x} {(- 6)}^{2} + {(- 6)}^{3} + 18 {(\sqrt[3]{x} - 6)}^{2} + 108 (\sqrt[3]{x} - 6) + 216$

Step 4.2.3.2.3

Multiply $- 6$ by $3$ .

$f^{- 1} (\sqrt[3]{x} - 6) = x - 18 \sqrt[3]{x^{2}} + 3 \sqrt[3]{x} {(- 6)}^{2} + {(- 6)}^{3} + 18 {(\sqrt[3]{x} - 6)}^{2} + 108 (\sqrt[3]{x} - 6) + 216$

Step 4.2.3.2.4

Raise $- 6$ to the power of $2$ .

$f^{- 1} (\sqrt[3]{x} - 6) = x - 18 \sqrt[3]{x^{2}} + 3 \sqrt[3]{x} \cdot 36 + {(- 6)}^{3} + 18 {(\sqrt[3]{x} - 6)}^{2} + 108 (\sqrt[3]{x} - 6) + 216$

Step 4.2.3.2.5

Multiply $36$ by $3$ .

$f^{- 1} (\sqrt[3]{x} - 6) = x - 18 \sqrt[3]{x^{2}} + 108 \sqrt[3]{x} + {(- 6)}^{3} + 18 {(\sqrt[3]{x} - 6)}^{2} + 108 (\sqrt[3]{x} - 6) + 216$

Step 4.2.3.2.6

Raise $- 6$ to the power of $3$ .

$f^{- 1} (\sqrt[3]{x} - 6) = x - 18 \sqrt[3]{x^{2}} + 108 \sqrt[3]{x} - 216 + 18 {(\sqrt[3]{x} - 6)}^{2} + 108 (\sqrt[3]{x} - 6) + 216$

Step 4.2.3.3

Rewrite ${(\sqrt[3]{x} - 6)}^{2}$ as $(\sqrt[3]{x} - 6) (\sqrt[3]{x} - 6)$ .

$f^{- 1} (\sqrt[3]{x} - 6) = x - 18 \sqrt[3]{x^{2}} + 108 \sqrt[3]{x} - 216 + 18 ((\sqrt[3]{x} - 6) (\sqrt[3]{x} - 6)) + 108 (\sqrt[3]{x} - 6) + 216$

Step 4.2.3.4

Expand $(\sqrt[3]{x} - 6) (\sqrt[3]{x} - 6)$ using the FOIL Method.

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Step 4.2.3.4.1

Apply the distributive property.

$f^{- 1} (\sqrt[3]{x} - 6) = x - 18 \sqrt[3]{x^{2}} + 108 \sqrt[3]{x} - 216 + 18 (\sqrt[3]{x} (\sqrt[3]{x} - 6) - 6 (\sqrt[3]{x} - 6)) + 108 (\sqrt[3]{x} - 6) + 216$

Step 4.2.3.4.2

Apply the distributive property.

$f^{- 1} (\sqrt[3]{x} - 6) = x - 18 \sqrt[3]{x^{2}} + 108 \sqrt[3]{x} - 216 + 18 (\sqrt[3]{x} \sqrt[3]{x} + \sqrt[3]{x} \cdot - 6 - 6 (\sqrt[3]{x} - 6)) + 108 (\sqrt[3]{x} - 6) + 216$

Step 4.2.3.4.3

Apply the distributive property.

$f^{- 1} (\sqrt[3]{x} - 6) = x - 18 \sqrt[3]{x^{2}} + 108 \sqrt[3]{x} - 216 + 18 (\sqrt[3]{x} \sqrt[3]{x} + \sqrt[3]{x} \cdot - 6 - 6 \sqrt[3]{x} - 6 \cdot - 6) + 108 (\sqrt[3]{x} - 6) + 216$

Step 4.2.3.5

Simplify and combine like terms.

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Step 4.2.3.5.1

Simplify each term.

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Step 4.2.3.5.1.1

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

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Step 4.2.3.5.1.1.1

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

$f^{- 1} (\sqrt[3]{x} - 6) = x - 18 \sqrt[3]{x^{2}} + 108 \sqrt[3]{x} - 216 + 18 (\sqrt[3]{x} \sqrt[3]{x} + \sqrt[3]{x} \cdot - 6 - 6 \sqrt[3]{x} - 6 \cdot - 6) + 108 (\sqrt[3]{x} - 6) + 216$

Step 4.2.3.5.1.1.2

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

$f^{- 1} (\sqrt[3]{x} - 6) = x - 18 \sqrt[3]{x^{2}} + 108 \sqrt[3]{x} - 216 + 18 (\sqrt[3]{x} \sqrt[3]{x} + \sqrt[3]{x} \cdot - 6 - 6 \sqrt[3]{x} - 6 \cdot - 6) + 108 (\sqrt[3]{x} - 6) + 216$

Step 4.2.3.5.1.1.3

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

$f^{- 1} (\sqrt[3]{x} - 6) = x - 18 \sqrt[3]{x^{2}} + 108 \sqrt[3]{x} - 216 + 18 ({\sqrt[3]{x}}^{1 + 1} + \sqrt[3]{x} \cdot - 6 - 6 \sqrt[3]{x} - 6 \cdot - 6) + 108 (\sqrt[3]{x} - 6) + 216$

Step 4.2.3.5.1.1.4

Add $1$ and $1$ .

$f^{- 1} (\sqrt[3]{x} - 6) = x - 18 \sqrt[3]{x^{2}} + 108 \sqrt[3]{x} - 216 + 18 ({\sqrt[3]{x}}^{2} + \sqrt[3]{x} \cdot - 6 - 6 \sqrt[3]{x} - 6 \cdot - 6) + 108 (\sqrt[3]{x} - 6) + 216$

Step 4.2.3.5.1.2

Rewrite ${\sqrt[3]{x}}^{2}$ as $\sqrt[3]{x^{2}}$ .

$f^{- 1} (\sqrt[3]{x} - 6) = x - 18 \sqrt[3]{x^{2}} + 108 \sqrt[3]{x} - 216 + 18 (\sqrt[3]{x^{2}} + \sqrt[3]{x} \cdot - 6 - 6 \sqrt[3]{x} - 6 \cdot - 6) + 108 (\sqrt[3]{x} - 6) + 216$

Step 4.2.3.5.1.3

Move $- 6$ to the left of $\sqrt[3]{x}$ .

$f^{- 1} (\sqrt[3]{x} - 6) = x - 18 \sqrt[3]{x^{2}} + 108 \sqrt[3]{x} - 216 + 18 (\sqrt[3]{x^{2}} - 6 \cdot \sqrt[3]{x} - 6 \sqrt[3]{x} - 6 \cdot - 6) + 108 (\sqrt[3]{x} - 6) + 216$

Step 4.2.3.5.1.4

Multiply $- 6$ by $- 6$ .

$f^{- 1} (\sqrt[3]{x} - 6) = x - 18 \sqrt[3]{x^{2}} + 108 \sqrt[3]{x} - 216 + 18 (\sqrt[3]{x^{2}} - 6 \sqrt[3]{x} - 6 \sqrt[3]{x} + 36) + 108 (\sqrt[3]{x} - 6) + 216$

Step 4.2.3.5.2

Subtract $6 \sqrt[3]{x}$ from $- 6 \sqrt[3]{x}$ .

$f^{- 1} (\sqrt[3]{x} - 6) = x - 18 \sqrt[3]{x^{2}} + 108 \sqrt[3]{x} - 216 + 18 (\sqrt[3]{x^{2}} - 12 \sqrt[3]{x} + 36) + 108 (\sqrt[3]{x} - 6) + 216$

Step 4.2.3.6

Apply the distributive property.

$f^{- 1} (\sqrt[3]{x} - 6) = x - 18 \sqrt[3]{x^{2}} + 108 \sqrt[3]{x} - 216 + 18 \sqrt[3]{x^{2}} + 18 (- 12 \sqrt[3]{x}) + 18 \cdot 36 + 108 (\sqrt[3]{x} - 6) + 216$

Step 4.2.3.7

Simplify.

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Step 4.2.3.7.1

Multiply $- 12$ by $18$ .

$f^{- 1} (\sqrt[3]{x} - 6) = x - 18 \sqrt[3]{x^{2}} + 108 \sqrt[3]{x} - 216 + 18 \sqrt[3]{x^{2}} - 216 \sqrt[3]{x} + 18 \cdot 36 + 108 (\sqrt[3]{x} - 6) + 216$

Step 4.2.3.7.2

Multiply $18$ by $36$ .

$f^{- 1} (\sqrt[3]{x} - 6) = x - 18 \sqrt[3]{x^{2}} + 108 \sqrt[3]{x} - 216 + 18 \sqrt[3]{x^{2}} - 216 \sqrt[3]{x} + 648 + 108 (\sqrt[3]{x} - 6) + 216$

Step 4.2.3.8

Apply the distributive property.

$f^{- 1} (\sqrt[3]{x} - 6) = x - 18 \sqrt[3]{x^{2}} + 108 \sqrt[3]{x} - 216 + 18 \sqrt[3]{x^{2}} - 216 \sqrt[3]{x} + 648 + 108 \sqrt[3]{x} + 108 \cdot - 6 + 216$

Step 4.2.3.9

Multiply $108$ by $- 6$ .

$f^{- 1} (\sqrt[3]{x} - 6) = x - 18 \sqrt[3]{x^{2}} + 108 \sqrt[3]{x} - 216 + 18 \sqrt[3]{x^{2}} - 216 \sqrt[3]{x} + 648 + 108 \sqrt[3]{x} - 648 + 216$

Step 4.2.4

Simplify by adding terms.

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Step 4.2.4.1

Combine the opposite terms in $x - 18 \sqrt[3]{x^{2}} + 108 \sqrt[3]{x} - 216 + 18 \sqrt[3]{x^{2}} - 216 \sqrt[3]{x} + 648 + 108 \sqrt[3]{x} - 648 + 216$ .

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Step 4.2.4.1.1

Add $- 18 \sqrt[3]{x^{2}}$ and $18 \sqrt[3]{x^{2}}$ .

$f^{- 1} (\sqrt[3]{x} - 6) = x + 0 + 108 \sqrt[3]{x} - 216 - 216 \sqrt[3]{x} + 648 + 108 \sqrt[3]{x} - 648 + 216$

Step 4.2.4.1.2

Add $x$ and $0$ .

$f^{- 1} (\sqrt[3]{x} - 6) = x + 108 \sqrt[3]{x} - 216 - 216 \sqrt[3]{x} + 648 + 108 \sqrt[3]{x} - 648 + 216$

Step 4.2.4.1.3

Subtract $648$ from $648$ .

$f^{- 1} (\sqrt[3]{x} - 6) = x + 108 \sqrt[3]{x} - 216 - 216 \sqrt[3]{x} + 0 + 108 \sqrt[3]{x} + 216$

Step 4.2.4.1.4

Add $x + 108 \sqrt[3]{x} - 216 - 216 \sqrt[3]{x}$ and $0$ .

$f^{- 1} (\sqrt[3]{x} - 6) = x + 108 \sqrt[3]{x} - 216 - 216 \sqrt[3]{x} + 108 \sqrt[3]{x} + 216$

Step 4.2.4.1.5

Add $- 216$ and $216$ .

$f^{- 1} (\sqrt[3]{x} - 6) = x + 108 \sqrt[3]{x} + 0 - 216 \sqrt[3]{x} + 108 \sqrt[3]{x}$

Step 4.2.4.1.6

Add $x + 108 \sqrt[3]{x}$ and $0$ .

$f^{- 1} (\sqrt[3]{x} - 6) = x + 108 \sqrt[3]{x} - 216 \sqrt[3]{x} + 108 \sqrt[3]{x}$

Step 4.2.4.2

Subtract $216 \sqrt[3]{x}$ from $108 \sqrt[3]{x}$ .

$f^{- 1} (\sqrt[3]{x} - 6) = x - 108 \sqrt[3]{x} + 108 \sqrt[3]{x}$

Step 4.2.4.3

Combine the opposite terms in $x - 108 \sqrt[3]{x} + 108 \sqrt[3]{x}$ .

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Step 4.2.4.3.1

Add $- 108 \sqrt[3]{x}$ and $108 \sqrt[3]{x}$ .

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

Step 4.2.4.3.2

Add $x$ and $0$ .

$f^{- 1} (\sqrt[3]{x} - 6) = x$

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 $f (x^{3} + 18 x^{2} + 108 x + 216)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (x^{3} + 18 x^{2} + 108 x + 216) = \sqrt[3]{x^{3} + 18 x^{2} + 108 x + 216} - 6$

Step 4.3.3

Remove parentheses.

$f (x^{3} + 18 x^{2} + 108 x + 216) = \sqrt[3]{x^{3} + 18 x^{2} + 108 x + 216} - 6$

Step 4.3.4

Simplify each term.

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Step 4.3.4.1

Make each term match the terms from the binomial theorem formula.

$f (x^{3} + 18 x^{2} + 108 x + 216) = \sqrt[3]{x^{3} + 3 \cdot (6 x^{2}) + 3 \cdot (6^{2} x) + 6^{3}} - 6$

Step 4.3.4.2

Factor $x^{3} + 3 \cdot 6 x^{2} + 3 \cdot 6^{2} x + 6^{3}$ using the binomial theorem.

$f (x^{3} + 18 x^{2} + 108 x + 216) = \sqrt[3]{{(x + 6)}^{3}} - 6$

Step 4.3.4.3

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

$f (x^{3} + 18 x^{2} + 108 x + 216) = x + 6 - 6$

Step 4.3.5

Combine the opposite terms in $x + 6 - 6$ .

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Step 4.3.5.1

Subtract $6$ from $6$ .

$f (x^{3} + 18 x^{2} + 108 x + 216) = x + 0$

Step 4.3.5.2

Add $x$ and $0$ .

$f (x^{3} + 18 x^{2} + 108 x + 216) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = x^{3} + 18 x^{2} + 108 x + 216$ is the inverse of $f (x) = \sqrt[3]{x} - 6$ .

$f^{- 1} (x) = x^{3} + 18 x^{2} + 108 x + 216$