Find the Inverse f(x)=(-6- cube root of 4x)/2

$f (x) = \frac{- 6 - \sqrt[3]{4 x}}{2}$

Step 1

Write $f (x) = \frac{- 6 - \sqrt[3]{4 x}}{2}$ as an equation.

$y = \frac{- 6 - \sqrt[3]{4 x}}{2}$

Step 2

Interchange the variables.

$x = \frac{- 6 - \sqrt[3]{4 y}}{2}$

Step 3

Solve for $y$ .

Rewrite the equation as $\frac{- 6 - \sqrt[3]{4 y}}{2} = x$ .

$\frac{- 6 - \sqrt[3]{4 y}}{2} = x$

Multiply both sides by $2$ .

$\frac{- 6 - \sqrt[3]{4 y}}{2} \cdot 2 = x \cdot 2$

Simplify.

Simplify the left side.

Simplify $\frac{- 6 - \sqrt[3]{4 y}}{2} \cdot 2$ .

Factor $- 1$ out of $- 6 - \sqrt[3]{4 y}$ .

Rewrite $- 6$ as $- 1 (6)$ .

$\frac{- 1 (6) - \sqrt[3]{4 y}}{2} \cdot 2 = x \cdot 2$

Factor $- 1$ out of $- \sqrt[3]{4 y}$ .

$\frac{- 1 (6) - (\sqrt[3]{4 y})}{2} \cdot 2 = x \cdot 2$

Factor $- 1$ out of $- 1 (6) - (\sqrt[3]{4 y})$ .

$\frac{- 1 (6 + \sqrt[3]{4 y})}{2} \cdot 2 = x \cdot 2$

Rewrite $- 1 (6 + \sqrt[3]{4 y})$ as $- (6 + \sqrt[3]{4 y})$ .

$\frac{- (6 + \sqrt[3]{4 y})}{2} \cdot 2 = x \cdot 2$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{- (6 + \sqrt[3]{4 y})}{2} \cdot 2 = x \cdot 2$

Rewrite the expression.

$- (6 + \sqrt[3]{4 y}) = x \cdot 2$

Apply the distributive property.

$- 1 \cdot 6 - \sqrt[3]{4 y} = x \cdot 2$

Multiply $- 1$ by $6$ .

$- 6 - \sqrt[3]{4 y} = x \cdot 2$

Simplify the right side.

Move $2$ to the left of $x$ .

$- 6 - \sqrt[3]{4 y} = 2 x$

Solve for $y$ .

Add $6$ to both sides of the equation.

$- \sqrt[3]{4 y} = 2 x + 6$

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

${(- \sqrt[3]{4 y})}^{3} = {(2 x + 6)}^{3}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4 y}$ as ${(4 y)}^{\frac{1}{3}}$ .

${(- {(4 y)}^{\frac{1}{3}})}^{3} = {(2 x + 6)}^{3}$

Simplify the left side.

Simplify ${(- {(4 y)}^{\frac{1}{3}})}^{3}$ .

Apply the product rule to $4 y$ .

${(- (4^{\frac{1}{3}} y^{\frac{1}{3}}))}^{3} = {(2 x + 6)}^{3}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- 4^{\frac{1}{3}} y^{\frac{1}{3}}$ .

${(- 4^{\frac{1}{3}})}^{3} {(y^{\frac{1}{3}})}^{3} = {(2 x + 6)}^{3}$

Apply the product rule to $- 4^{\frac{1}{3}}$ .

${(- 1)}^{3} {(4^{\frac{1}{3}})}^{3} {(y^{\frac{1}{3}})}^{3} = {(2 x + 6)}^{3}$

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

$- {(4^{\frac{1}{3}})}^{3} {(y^{\frac{1}{3}})}^{3} = {(2 x + 6)}^{3}$

Multiply the exponents in ${(4^{\frac{1}{3}})}^{3}$ .

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

$- 4^{\frac{1}{3} \cdot 3} {(y^{\frac{1}{3}})}^{3} = {(2 x + 6)}^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$- 4^{\frac{1}{3} \cdot 3} {(y^{\frac{1}{3}})}^{3} = {(2 x + 6)}^{3}$

Rewrite the expression.

$- 4^{1} {(y^{\frac{1}{3}})}^{3} = {(2 x + 6)}^{3}$

Evaluate the exponent.

$- 1 \cdot 4 {(y^{\frac{1}{3}})}^{3} = {(2 x + 6)}^{3}$

Multiply $- 1$ by $4$ .

$- 4 {(y^{\frac{1}{3}})}^{3} = {(2 x + 6)}^{3}$

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

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

$- 4 y^{\frac{1}{3} \cdot 3} = {(2 x + 6)}^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$- 4 y^{\frac{1}{3} \cdot 3} = {(2 x + 6)}^{3}$

Rewrite the expression.

$- 4 y^{1} = {(2 x + 6)}^{3}$

Simplify.

$- 4 y = {(2 x + 6)}^{3}$

Simplify the right side.

Simplify ${(2 x + 6)}^{3}$ .

Use the Binomial Theorem.

$- 4 y = {(2 x)}^{3} + 3 {(2 x)}^{2} \cdot 6 + 3 (2 x) \cdot 6^{2} + 6^{3}$

Simplify each term.

Apply the product rule to $2 x$ .

$- 4 y = 2^{3} x^{3} + 3 {(2 x)}^{2} \cdot 6 + 3 (2 x) \cdot 6^{2} + 6^{3}$

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

$- 4 y = 8 x^{3} + 3 {(2 x)}^{2} \cdot 6 + 3 (2 x) \cdot 6^{2} + 6^{3}$

Apply the product rule to $2 x$ .

$- 4 y = 8 x^{3} + 3 (2^{2} x^{2}) \cdot 6 + 3 (2 x) \cdot 6^{2} + 6^{3}$

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

$- 4 y = 8 x^{3} + 3 (4 x^{2}) \cdot 6 + 3 (2 x) \cdot 6^{2} + 6^{3}$

Multiply $4$ by $3$ .

$- 4 y = 8 x^{3} + 12 x^{2} \cdot 6 + 3 (2 x) \cdot 6^{2} + 6^{3}$

Multiply $6$ by $12$ .

$- 4 y = 8 x^{3} + 72 x^{2} + 3 (2 x) \cdot 6^{2} + 6^{3}$

Multiply $2$ by $3$ .

$- 4 y = 8 x^{3} + 72 x^{2} + 6 x \cdot 6^{2} + 6^{3}$

Multiply $6$ by $6^{2}$ by adding the exponents.

Move $6^{2}$ .

$- 4 y = 8 x^{3} + 72 x^{2} + 6^{2} \cdot 6 x + 6^{3}$

Multiply $6^{2}$ by $6$ .

Raise $6$ to the power of $1$ .

$- 4 y = 8 x^{3} + 72 x^{2} + 6^{2} \cdot 6^{1} x + 6^{3}$

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

$- 4 y = 8 x^{3} + 72 x^{2} + 6^{2 + 1} x + 6^{3}$

Add $2$ and $1$ .

$- 4 y = 8 x^{3} + 72 x^{2} + 6^{3} x + 6^{3}$

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

$- 4 y = 8 x^{3} + 72 x^{2} + 216 x + 6^{3}$

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

$- 4 y = 8 x^{3} + 72 x^{2} + 216 x + 216$

Divide each term in $- 4 y = 8 x^{3} + 72 x^{2} + 216 x + 216$ by $- 4$ and simplify.

Divide each term in $- 4 y = 8 x^{3} + 72 x^{2} + 216 x + 216$ by $- 4$ .

$\frac{- 4 y}{- 4} = \frac{8 x^{3}}{- 4} + \frac{72 x^{2}}{- 4} + \frac{216 x}{- 4} + \frac{216}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 y}{- 4} = \frac{8 x^{3}}{- 4} + \frac{72 x^{2}}{- 4} + \frac{216 x}{- 4} + \frac{216}{- 4}$

Divide $y$ by $1$ .

$y = \frac{8 x^{3}}{- 4} + \frac{72 x^{2}}{- 4} + \frac{216 x}{- 4} + \frac{216}{- 4}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $8$ and $- 4$ .

Factor $4$ out of $8 x^{3}$ .

$y = \frac{4 (2 x^{3})}{- 4} + \frac{72 x^{2}}{- 4} + \frac{216 x}{- 4} + \frac{216}{- 4}$

Move the negative one from the denominator of $\frac{2 x^{3}}{- 1}$ .

$y = - 1 \cdot (2 x^{3}) + \frac{72 x^{2}}{- 4} + \frac{216 x}{- 4} + \frac{216}{- 4}$

Rewrite $- 1 \cdot (2 x^{3})$ as $- (2 x^{3})$ .

$y = - (2 x^{3}) + \frac{72 x^{2}}{- 4} + \frac{216 x}{- 4} + \frac{216}{- 4}$

Multiply $2$ by $- 1$ .

$y = - 2 x^{3} + \frac{72 x^{2}}{- 4} + \frac{216 x}{- 4} + \frac{216}{- 4}$

Cancel the common factor of $72$ and $- 4$ .

Factor $4$ out of $72 x^{2}$ .

$y = - 2 x^{3} + \frac{4 (18 x^{2})}{- 4} + \frac{216 x}{- 4} + \frac{216}{- 4}$

Move the negative one from the denominator of $\frac{18 x^{2}}{- 1}$ .

$y = - 2 x^{3} - 1 \cdot (18 x^{2}) + \frac{216 x}{- 4} + \frac{216}{- 4}$

Rewrite $- 1 \cdot (18 x^{2})$ as $- (18 x^{2})$ .

$y = - 2 x^{3} - (18 x^{2}) + \frac{216 x}{- 4} + \frac{216}{- 4}$

Multiply $18$ by $- 1$ .

$y = - 2 x^{3} - 18 x^{2} + \frac{216 x}{- 4} + \frac{216}{- 4}$

Cancel the common factor of $216$ and $- 4$ .

Factor $4$ out of $216 x$ .

$y = - 2 x^{3} - 18 x^{2} + \frac{4 (54 x)}{- 4} + \frac{216}{- 4}$

Move the negative one from the denominator of $\frac{54 x}{- 1}$ .

$y = - 2 x^{3} - 18 x^{2} - 1 \cdot (54 x) + \frac{216}{- 4}$

Rewrite $- 1 \cdot (54 x)$ as $- (54 x)$ .

$y = - 2 x^{3} - 18 x^{2} - (54 x) + \frac{216}{- 4}$

Multiply $54$ by $- 1$ .

$y = - 2 x^{3} - 18 x^{2} - 54 x + \frac{216}{- 4}$

Divide $216$ by $- 4$ .

$y = - 2 x^{3} - 18 x^{2} - 54 x - 54$

Step 4

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

$f^{- 1} (x) = - 2 x^{3} - 18 x^{2} - 54 x - 54$

Step 5

Verify if $f^{- 1} (x) = - 2 x^{3} - 18 x^{2} - 54 x - 54$ is the inverse of $f (x) = \frac{- 6 - \sqrt[3]{4 x}}{2}$ .

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

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

Set up the composite result function.

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

Evaluate $f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 2 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{3} - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Simplify each term.

Factor $- 1$ out of $- 6 - \sqrt[3]{4 x}$ .

Rewrite $- 6$ as $- 1 (6)$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 2 {(\frac{- 1 \cdot 6 - \sqrt[3]{4 x}}{2})}^{3} - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Factor $- 1$ out of $- \sqrt[3]{4 x}$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 2 {(\frac{- 1 \cdot 6 - (\sqrt[3]{4 x})}{2})}^{3} - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Factor $- 1$ out of $- 1 (6) - (\sqrt[3]{4 x})$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 2 {(\frac{- 1 (6 + \sqrt[3]{4 x})}{2})}^{3} - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Rewrite $- 1 (6 + \sqrt[3]{4 x})$ as $- (6 + \sqrt[3]{4 x})$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 2 {(\frac{- (6 + \sqrt[3]{4 x})}{2})}^{3} - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Move the negative in front of the fraction.

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 2 {(- \frac{6 + \sqrt[3]{4 x}}{2})}^{3} - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{6 + \sqrt[3]{4 x}}{2}$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 2 ({(- 1)}^{3} {(\frac{6 + \sqrt[3]{4 x}}{2})}^{3}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Apply the product rule to $\frac{6 + \sqrt[3]{4 x}}{2}$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 2 ({(- 1)}^{3} (\frac{{(6 + \sqrt[3]{4 x})}^{3}}{2^{3}})) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

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

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 2 (- \frac{{(6 + \sqrt[3]{4 x})}^{3}}{2^{3}}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

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

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 2 (- \frac{{(6 + \sqrt[3]{4 x})}^{3}}{8}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{{(6 + \sqrt[3]{4 x})}^{3}}{8}$ into the numerator.

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 2 \frac{- {(6 + \sqrt[3]{4 x})}^{3}}{8} - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Factor $2$ out of $- 2$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = 2 (- 1) (\frac{- {(6 + \sqrt[3]{4 x})}^{3}}{8}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Factor $2$ out of $8$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = 2 \cdot (- 1 \frac{- {(6 + \sqrt[3]{4 x})}^{3}}{2 \cdot 4}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Cancel the common factor.

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = 2 \cdot (- 1 \frac{- {(6 + \sqrt[3]{4 x})}^{3}}{2 \cdot 4}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Rewrite the expression.

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 \frac{- {(6 + \sqrt[3]{4 x})}^{3}}{4} - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Move the negative in front of the fraction.

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- \frac{{(6 + \sqrt[3]{4 x})}^{3}}{4}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Simplify ${(6 + \sqrt[3]{4 x})}^{3}$ .

Use the Binomial Theorem.

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- \frac{6^{3} + 3 \cdot (6^{2} \sqrt[3]{4 x}) + 3 \cdot (6 {\sqrt[3]{4 x}}^{2}) + {\sqrt[3]{4 x}}^{3}}{4}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Simplify each term.

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

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- \frac{216 + 3 \cdot (6^{2} \sqrt[3]{4 x}) + 3 \cdot (6 {\sqrt[3]{4 x}}^{2}) + {\sqrt[3]{4 x}}^{3}}{4}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

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

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- \frac{216 + 3 \cdot (36 \sqrt[3]{4 x}) + 3 \cdot (6 {\sqrt[3]{4 x}}^{2}) + {\sqrt[3]{4 x}}^{3}}{4}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Multiply $3$ by $36$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- \frac{216 + 108 \sqrt[3]{4 x} + 3 \cdot (6 {\sqrt[3]{4 x}}^{2}) + {\sqrt[3]{4 x}}^{3}}{4}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Multiply $3$ by $6$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- \frac{216 + 108 \sqrt[3]{4 x} + 18 {\sqrt[3]{4 x}}^{2} + {\sqrt[3]{4 x}}^{3}}{4}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Rewrite ${\sqrt[3]{4 x}}^{2}$ as ${({(4 x)}^{2})}^{\frac{1}{3}}$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- \frac{216 + 108 \sqrt[3]{4 x} + 18 \sqrt[3]{{(4 x)}^{2}} + {\sqrt[3]{4 x}}^{3}}{4}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Apply the product rule to $4 x$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- \frac{216 + 108 \sqrt[3]{4 x} + 18 \sqrt[3]{4^{2} x^{2}} + {\sqrt[3]{4 x}}^{3}}{4}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Raise $4$ to the power of $2$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- \frac{216 + 108 \sqrt[3]{4 x} + 18 \sqrt[3]{16 x^{2}} + {\sqrt[3]{4 x}}^{3}}{4}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Rewrite $16 x^{2}$ as $2^{3} \cdot (2 x^{2})$ .

Factor $8$ out of $16$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- \frac{216 + 108 \sqrt[3]{4 x} + 18 \sqrt[3]{8 (2) x^{2}} + {\sqrt[3]{4 x}}^{3}}{4}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Rewrite $8$ as $2^{3}$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- \frac{216 + 108 \sqrt[3]{4 x} + 18 \sqrt[3]{2^{3} \cdot (2 x^{2})} + {\sqrt[3]{4 x}}^{3}}{4}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Add parentheses.

Pull terms out from under the radical.

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- \frac{216 + 108 \sqrt[3]{4 x} + 18 (2 \sqrt[3]{2 x^{2}}) + {\sqrt[3]{4 x}}^{3}}{4}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Multiply $2$ by $18$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- \frac{216 + 108 \sqrt[3]{4 x} + 36 \sqrt[3]{2 x^{2}} + {\sqrt[3]{4 x}}^{3}}{4}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Rewrite ${\sqrt[3]{4 x}}^{3}$ as $4 x$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4 x}$ as ${(4 x)}^{\frac{1}{3}}$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- \frac{216 + 108 \sqrt[3]{4 x} + 36 \sqrt[3]{2 x^{2}} + {({(4 x)}^{\frac{1}{3}})}^{3}}{4}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

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

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- \frac{216 + 108 \sqrt[3]{4 x} + 36 \sqrt[3]{2 x^{2}} + {(4 x)}^{\frac{1}{3} \cdot 3}}{4}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

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

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- \frac{216 + 108 \sqrt[3]{4 x} + 36 \sqrt[3]{2 x^{2}} + {(4 x)}^{\frac{3}{3}}}{4}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Cancel the common factor of $3$ .

Cancel the common factor.

Rewrite the expression.

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- \frac{216 + 108 \sqrt[3]{4 x} + 36 \sqrt[3]{2 x^{2}} + 4 x}{4}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Simplify.

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- \frac{216 + 108 \sqrt[3]{4 x} + 36 \sqrt[3]{2 x^{2}} + 4 x}{4}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Move $36 \sqrt[3]{2 x^{2}}$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- \frac{216 + 108 \sqrt[3]{4 x} + 4 x + 36 \sqrt[3]{2 x^{2}}}{4}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Move $108 \sqrt[3]{4 x}$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- \frac{216 + 4 x + 108 \sqrt[3]{4 x} + 36 \sqrt[3]{2 x^{2}}}{4}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Reorder $216$ and $4 x$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- \frac{4 x + 216 + 108 \sqrt[3]{4 x} + 36 \sqrt[3]{2 x^{2}}}{4}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Cancel the common factor of $4 x + 216 + 108 \sqrt[3]{4 x} + 36 \sqrt[3]{2 x^{2}}$ and $4$ .

Factor $4$ out of $4 x$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- \frac{4 (x) + 216 + 108 \sqrt[3]{4 x} + 36 \sqrt[3]{2 x^{2}}}{4}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Factor $4$ out of $216$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- \frac{4 (x) + 4 \cdot 54 + 108 \sqrt[3]{4 x} + 36 \sqrt[3]{2 x^{2}}}{4}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Factor $4$ out of $4 (x) + 4 (54)$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- \frac{4 (x + 54) + 108 \sqrt[3]{4 x} + 36 \sqrt[3]{2 x^{2}}}{4}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Factor $4$ out of $108 \sqrt[3]{4 x}$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- \frac{4 (x + 54) + 4 (27 \sqrt[3]{4 x}) + 36 \sqrt[3]{2 x^{2}}}{4}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Factor $4$ out of $4 (x + 54) + 4 (27 \sqrt[3]{4 x})$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- \frac{4 (x + 54 + 27 \sqrt[3]{4 x}) + 36 \sqrt[3]{2 x^{2}}}{4}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Factor $4$ out of $36 \sqrt[3]{2 x^{2}}$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- \frac{4 (x + 54 + 27 \sqrt[3]{4 x}) + 4 (9 \sqrt[3]{2 x^{2}})}{4}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Factor $4$ out of $4 (x + 54 + 27 \sqrt[3]{4 x}) + 4 (9 \sqrt[3]{2 x^{2}})$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- \frac{4 (x + 54 + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}})}{4}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Cancel the common factors.

Factor $4$ out of $4$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- \frac{4 (x + 54 + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}})}{4 (1)}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Cancel the common factor.

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- \frac{4 (x + 54 + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}})}{4 \cdot 1}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Rewrite the expression.

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- \frac{x + 54 + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}}}{1}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Divide $x + 54 + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}}$ by $1$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- (x + 54 + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}})) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Apply the distributive property.

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- x - 1 \cdot 54 - (27 \sqrt[3]{4 x}) - (9 \sqrt[3]{2 x^{2}})) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Simplify.

Multiply $- 1$ by $54$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- x - 54 - (27 \sqrt[3]{4 x}) - (9 \sqrt[3]{2 x^{2}})) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Multiply $27$ by $- 1$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- x - 54 - 27 \sqrt[3]{4 x} - (9 \sqrt[3]{2 x^{2}})) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Multiply $9$ by $- 1$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- x - 54 - 27 \sqrt[3]{4 x} - 9 \sqrt[3]{2 x^{2}}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Apply the distributive property.

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = - 1 (- x) - 1 \cdot - 54 - 1 (- 27 \sqrt[3]{4 x}) - 1 (- 9 \sqrt[3]{2 x^{2}}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Simplify.

Multiply $- 1 (- x)$ .

Multiply $- 1$ by $- 1$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = 1 x - 1 \cdot - 54 - 1 (- 27 \sqrt[3]{4 x}) - 1 (- 9 \sqrt[3]{2 x^{2}}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Multiply $x$ by $1$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x - 1 \cdot - 54 - 1 (- 27 \sqrt[3]{4 x}) - 1 (- 9 \sqrt[3]{2 x^{2}}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Multiply $- 1$ by $- 54$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x + 54 - 1 (- 27 \sqrt[3]{4 x}) - 1 (- 9 \sqrt[3]{2 x^{2}}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Multiply $- 27$ by $- 1$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x + 54 + 27 \sqrt[3]{4 x} - 1 (- 9 \sqrt[3]{2 x^{2}}) - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Multiply $- 9$ by $- 1$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x + 54 + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} - 18 {(\frac{- 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Factor $- 1$ out of $- 6 - \sqrt[3]{4 x}$ .

Rewrite $- 6$ as $- 1 (6)$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x + 54 + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} - 18 {(\frac{- 1 \cdot 6 - \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Factor $- 1$ out of $- \sqrt[3]{4 x}$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x + 54 + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} - 18 {(\frac{- 1 \cdot 6 - (\sqrt[3]{4 x})}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Factor $- 1$ out of $- 1 (6) - (\sqrt[3]{4 x})$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x + 54 + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} - 18 {(\frac{- 1 (6 + \sqrt[3]{4 x})}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Rewrite $- 1 (6 + \sqrt[3]{4 x})$ as $- (6 + \sqrt[3]{4 x})$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x + 54 + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} - 18 {(\frac{- (6 + \sqrt[3]{4 x})}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Move the negative in front of the fraction.

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x + 54 + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} - 18 {(- \frac{6 + \sqrt[3]{4 x}}{2})}^{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{6 + \sqrt[3]{4 x}}{2}$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x + 54 + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} - 18 ({(- 1)}^{2} {(\frac{6 + \sqrt[3]{4 x}}{2})}^{2}) - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Apply the product rule to $\frac{6 + \sqrt[3]{4 x}}{2}$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x + 54 + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} - 18 ({(- 1)}^{2} (\frac{{(6 + \sqrt[3]{4 x})}^{2}}{2^{2}})) - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

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

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x + 54 + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} - 18 (1 (\frac{{(6 + \sqrt[3]{4 x})}^{2}}{2^{2}})) - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Multiply $\frac{{(6 + \sqrt[3]{4 x})}^{2}}{2^{2}}$ by $1$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x + 54 + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} - 18 \frac{{(6 + \sqrt[3]{4 x})}^{2}}{2^{2}} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

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

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x + 54 + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} - 18 \frac{{(6 + \sqrt[3]{4 x})}^{2}}{4} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Cancel the common factor of $2$ .

Factor $2$ out of $- 18$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x + 54 + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 2 (- 9) (\frac{{(6 + \sqrt[3]{4 x})}^{2}}{4}) - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Factor $2$ out of $4$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x + 54 + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 2 \cdot (- 9 \frac{{(6 + \sqrt[3]{4 x})}^{2}}{2 \cdot 2}) - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Cancel the common factor.

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x + 54 + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 2 \cdot (- 9 \frac{{(6 + \sqrt[3]{4 x})}^{2}}{2 \cdot 2}) - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Rewrite the expression.

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x + 54 + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} - 9 \frac{{(6 + \sqrt[3]{4 x})}^{2}}{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Combine $- 9$ and $\frac{{(6 + \sqrt[3]{4 x})}^{2}}{2}$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x + 54 + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + \frac{- 9 {(6 + \sqrt[3]{4 x})}^{2}}{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Move the negative in front of the fraction.

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x + 54 + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} - \frac{(9) {(6 + \sqrt[3]{4 x})}^{2}}{2} - 54 \frac{- 6 - \sqrt[3]{4 x}}{2} - 54$

Factor $- 1$ out of $- 6 - \sqrt[3]{4 x}$ .

Rewrite $- 6$ as $- 1 (6)$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x + 54 + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} - \frac{9 {(6 + \sqrt[3]{4 x})}^{2}}{2} - 54 \frac{- 1 \cdot 6 - \sqrt[3]{4 x}}{2} - 54$

Factor $- 1$ out of $- \sqrt[3]{4 x}$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x + 54 + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} - \frac{9 {(6 + \sqrt[3]{4 x})}^{2}}{2} - 54 \frac{- 1 \cdot 6 - (\sqrt[3]{4 x})}{2} - 54$

Factor $- 1$ out of $- 1 (6) - (\sqrt[3]{4 x})$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x + 54 + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} - \frac{9 {(6 + \sqrt[3]{4 x})}^{2}}{2} - 54 \frac{- 1 (6 + \sqrt[3]{4 x})}{2} - 54$

Rewrite $- 1 (6 + \sqrt[3]{4 x})$ as $- (6 + \sqrt[3]{4 x})$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x + 54 + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} - \frac{9 {(6 + \sqrt[3]{4 x})}^{2}}{2} - 54 \frac{- (6 + \sqrt[3]{4 x})}{2} - 54$

Cancel the common factor of $2$ .

Factor $2$ out of $- 54$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x + 54 + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} - \frac{9 {(6 + \sqrt[3]{4 x})}^{2}}{2} + 2 (- 27) (\frac{- (6 + \sqrt[3]{4 x})}{2}) - 54$

Cancel the common factor.

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x + 54 + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} - \frac{9 {(6 + \sqrt[3]{4 x})}^{2}}{2} + 2 \cdot (- 27 \frac{- (6 + \sqrt[3]{4 x})}{2}) - 54$

Rewrite the expression.

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x + 54 + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} - \frac{9 {(6 + \sqrt[3]{4 x})}^{2}}{2} - 27 (- (6 + \sqrt[3]{4 x})) - 54$

Multiply $- 1$ by $- 27$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x + 54 + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} - \frac{9 {(6 + \sqrt[3]{4 x})}^{2}}{2} + 27 (6 + \sqrt[3]{4 x}) - 54$

Apply the distributive property.

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x + 54 + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} - \frac{9 {(6 + \sqrt[3]{4 x})}^{2}}{2} + 27 \cdot 6 + 27 \sqrt[3]{4 x} - 54$

Multiply $27$ by $6$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x + 54 + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} - \frac{9 {(6 + \sqrt[3]{4 x})}^{2}}{2} + 162 + 27 \sqrt[3]{4 x} - 54$

Combine the opposite terms in $x + 54 + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} - \frac{9 {(6 + \sqrt[3]{4 x})}^{2}}{2} + 162 + 27 \sqrt[3]{4 x} - 54$ .

Subtract $54$ from $54$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x + 0 + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} - \frac{9 {(6 + \sqrt[3]{4 x})}^{2}}{2} + 162 + 27 \sqrt[3]{4 x}$

Add $x$ and $0$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} - \frac{9 {(6 + \sqrt[3]{4 x})}^{2}}{2} + 162 + 27 \sqrt[3]{4 x}$

To write $x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x \cdot \frac{2}{2} - \frac{9 {(6 + \sqrt[3]{4 x})}^{2}}{2} + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 162 + 27 \sqrt[3]{4 x}$

Simplify terms.

Combine $x$ and $\frac{2}{2}$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = \frac{x \cdot 2}{2} - \frac{9 {(6 + \sqrt[3]{4 x})}^{2}}{2} + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 162 + 27 \sqrt[3]{4 x}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = \frac{x \cdot 2 - 9 {(6 + \sqrt[3]{4 x})}^{2}}{2} + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 162 + 27 \sqrt[3]{4 x}$

Simplify each term.

Simplify the numerator.

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

Move $2$ to the left of $u$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = \frac{2 \cdot u - 9 {(6 + \sqrt[3]{4 u})}^{2}}{2} + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 162 + 27 \sqrt[3]{4 x}$

Rewrite ${(6 + \sqrt[3]{4 u})}^{2}$ as $(6 + \sqrt[3]{4 u}) (6 + \sqrt[3]{4 u})$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = \frac{2 u - 9 ((6 + \sqrt[3]{4 u}) (6 + \sqrt[3]{4 u}))}{2} + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 162 + 27 \sqrt[3]{4 x}$

Expand $(6 + \sqrt[3]{4 u}) (6 + \sqrt[3]{4 u})$ using the FOIL Method.

Apply the distributive property.

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = \frac{2 u - 9 (6 (6 + \sqrt[3]{4 u}) + \sqrt[3]{4 u} (6 + \sqrt[3]{4 u}))}{2} + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 162 + 27 \sqrt[3]{4 x}$

Apply the distributive property.

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = \frac{2 u - 9 (6 \cdot 6 + 6 \sqrt[3]{4 u} + \sqrt[3]{4 u} (6 + \sqrt[3]{4 u}))}{2} + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 162 + 27 \sqrt[3]{4 x}$

Apply the distributive property.

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = \frac{2 u - 9 (6 \cdot 6 + 6 \sqrt[3]{4 u} + \sqrt[3]{4 u} \cdot 6 + \sqrt[3]{4 u} \sqrt[3]{4 u})}{2} + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 162 + 27 \sqrt[3]{4 x}$

Simplify and combine like terms.

Simplify each term.

Multiply $6$ by $6$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = \frac{2 u - 9 (36 + 6 \sqrt[3]{4 u} + \sqrt[3]{4 u} \cdot 6 + \sqrt[3]{4 u} \sqrt[3]{4 u})}{2} + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 162 + 27 \sqrt[3]{4 x}$

Move $6$ to the left of $\sqrt[3]{4 u}$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = \frac{2 u - 9 (36 + 6 \sqrt[3]{4 u} + 6 \sqrt[3]{4 u} + \sqrt[3]{4 u} \sqrt[3]{4 u})}{2} + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 162 + 27 \sqrt[3]{4 x}$

Multiply $\sqrt[3]{4 u} \sqrt[3]{4 u}$ .

Raise $\sqrt[3]{4 u}$ to the power of $1$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = \frac{2 u - 9 (36 + 6 \sqrt[3]{4 u} + 6 \sqrt[3]{4 u} + \sqrt[3]{4 u} \sqrt[3]{4 u})}{2} + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 162 + 27 \sqrt[3]{4 x}$

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

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = \frac{2 u - 9 (36 + 6 \sqrt[3]{4 u} + 6 \sqrt[3]{4 u} + {\sqrt[3]{4 u}}^{1 + 1})}{2} + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 162 + 27 \sqrt[3]{4 x}$

Add $1$ and $1$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = \frac{2 u - 9 (36 + 6 \sqrt[3]{4 u} + 6 \sqrt[3]{4 u} + {\sqrt[3]{4 u}}^{2})}{2} + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 162 + 27 \sqrt[3]{4 x}$

Rewrite ${\sqrt[3]{4 u}}^{2}$ as ${({(4 u)}^{2})}^{\frac{1}{3}}$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = \frac{2 u - 9 (36 + 6 \sqrt[3]{4 u} + 6 \sqrt[3]{4 u} + \sqrt[3]{{(4 u)}^{2}})}{2} + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 162 + 27 \sqrt[3]{4 x}$

Apply the product rule to $4 u$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = \frac{2 u - 9 (36 + 6 \sqrt[3]{4 u} + 6 \sqrt[3]{4 u} + \sqrt[3]{4^{2} u^{2}})}{2} + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 162 + 27 \sqrt[3]{4 x}$

Raise $4$ to the power of $2$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = \frac{2 u - 9 (36 + 6 \sqrt[3]{4 u} + 6 \sqrt[3]{4 u} + \sqrt[3]{16 u^{2}})}{2} + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 162 + 27 \sqrt[3]{4 x}$

Rewrite $16 u^{2}$ as $2^{3} \cdot (2 u^{2})$ .

Factor $8$ out of $16$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = \frac{2 u - 9 (36 + 6 \sqrt[3]{4 u} + 6 \sqrt[3]{4 u} + \sqrt[3]{8 (2) u^{2}})}{2} + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 162 + 27 \sqrt[3]{4 x}$

Rewrite $8$ as $2^{3}$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = \frac{2 u - 9 (36 + 6 \sqrt[3]{4 u} + 6 \sqrt[3]{4 u} + \sqrt[3]{2^{3} \cdot (2 u^{2})})}{2} + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 162 + 27 \sqrt[3]{4 x}$

Add parentheses.

Pull terms out from under the radical.

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = \frac{2 u - 9 (36 + 6 \sqrt[3]{4 u} + 6 \sqrt[3]{4 u} + 2 \sqrt[3]{2 u^{2}})}{2} + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 162 + 27 \sqrt[3]{4 x}$

Add $6 \sqrt[3]{4 u}$ and $6 \sqrt[3]{4 u}$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = \frac{2 u - 9 (36 + 12 \sqrt[3]{4 u} + 2 \sqrt[3]{2 u^{2}})}{2} + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 162 + 27 \sqrt[3]{4 x}$

Apply the distributive property.

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = \frac{2 u - 9 \cdot 36 - 9 (12 \sqrt[3]{4 u}) - 9 (2 \sqrt[3]{2 u^{2}})}{2} + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 162 + 27 \sqrt[3]{4 x}$

Simplify.

Multiply $- 9$ by $36$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = \frac{2 u - 324 - 9 (12 \sqrt[3]{4 u}) - 9 (2 \sqrt[3]{2 u^{2}})}{2} + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 162 + 27 \sqrt[3]{4 x}$

Multiply $12$ by $- 9$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = \frac{2 u - 324 - 108 \sqrt[3]{4 u} - 9 (2 \sqrt[3]{2 u^{2}})}{2} + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 162 + 27 \sqrt[3]{4 x}$

Multiply $2$ by $- 9$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = \frac{2 u - 324 - 108 \sqrt[3]{4 u} - 18 \sqrt[3]{2 u^{2}}}{2} + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 162 + 27 \sqrt[3]{4 x}$

Factor $2$ out of $2 u - 324 - 108 \sqrt[3]{4 u} - 18 \sqrt[3]{2 u^{2}}$ .

Factor $2$ out of $2 u$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = \frac{2 (u) - 324 - 108 \sqrt[3]{4 u} - 18 \sqrt[3]{2 u^{2}}}{2} + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 162 + 27 \sqrt[3]{4 x}$

Factor $2$ out of $- 324$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = \frac{2 u + 2 \cdot - 162 - 108 \sqrt[3]{4 u} - 18 \sqrt[3]{2 u^{2}}}{2} + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 162 + 27 \sqrt[3]{4 x}$

Factor $2$ out of $- 108 \sqrt[3]{4 u}$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = \frac{2 u + 2 \cdot - 162 + 2 (- 54 \sqrt[3]{4 u}) - 18 \sqrt[3]{2 u^{2}}}{2} + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 162 + 27 \sqrt[3]{4 x}$

Factor $2$ out of $- 18 \sqrt[3]{2 u^{2}}$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = \frac{2 u + 2 \cdot - 162 + 2 (- 54 \sqrt[3]{4 u}) + 2 (- 9 \sqrt[3]{2 u^{2}})}{2} + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 162 + 27 \sqrt[3]{4 x}$

Factor $2$ out of $2 u + 2 \cdot - 162$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = \frac{2 (u - 162) + 2 (- 54 \sqrt[3]{4 u}) + 2 (- 9 \sqrt[3]{2 u^{2}})}{2} + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 162 + 27 \sqrt[3]{4 x}$

Factor $2$ out of $2 (u - 162) + 2 (- 54 \sqrt[3]{4 u})$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = \frac{2 (u - 162 - 54 \sqrt[3]{4 u}) + 2 (- 9 \sqrt[3]{2 u^{2}})}{2} + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 162 + 27 \sqrt[3]{4 x}$

Factor $2$ out of $2 (u - 162 - 54 \sqrt[3]{4 u}) + 2 (- 9 \sqrt[3]{2 u^{2}})$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = \frac{2 (u - 162 - 54 \sqrt[3]{4 u} - 9 \sqrt[3]{2 u^{2}})}{2} + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 162 + 27 \sqrt[3]{4 x}$

Replace all occurrences of $u$ with $x$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = \frac{2 (x - 162 - 54 \sqrt[3]{4 x} - 9 \sqrt[3]{2 x^{2}})}{2} + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 162 + 27 \sqrt[3]{4 x}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = \frac{2 (x - 162 - 54 \sqrt[3]{4 x} - 9 \sqrt[3]{2 x^{2}})}{2} + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 162 + 27 \sqrt[3]{4 x}$

Divide $x - 162 - 54 \sqrt[3]{4 x} - 9 \sqrt[3]{2 x^{2}}$ by $1$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x - 162 - 54 \sqrt[3]{4 x} - 9 \sqrt[3]{2 x^{2}} + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 162 + 27 \sqrt[3]{4 x}$

Simplify by adding terms.

Combine the opposite terms in $x - 162 - 54 \sqrt[3]{4 x} - 9 \sqrt[3]{2 x^{2}} + 27 \sqrt[3]{4 x} + 9 \sqrt[3]{2 x^{2}} + 162 + 27 \sqrt[3]{4 x}$ .

Add $- 9 \sqrt[3]{2 x^{2}}$ and $9 \sqrt[3]{2 x^{2}}$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x - 162 - 54 \sqrt[3]{4 x} + 0 + 27 \sqrt[3]{4 x} + 162 + 27 \sqrt[3]{4 x}$

Add $x - 162 - 54 \sqrt[3]{4 x}$ and $0$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x - 162 - 54 \sqrt[3]{4 x} + 27 \sqrt[3]{4 x} + 162 + 27 \sqrt[3]{4 x}$

Add $- 162$ and $162$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x + 0 - 54 \sqrt[3]{4 x} + 27 \sqrt[3]{4 x} + 27 \sqrt[3]{4 x}$

Add $x$ and $0$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x - 54 \sqrt[3]{4 x} + 27 \sqrt[3]{4 x} + 27 \sqrt[3]{4 x}$

Add $- 54 \sqrt[3]{4 x}$ and $27 \sqrt[3]{4 x}$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x - 27 \sqrt[3]{4 x} + 27 \sqrt[3]{4 x}$

Combine the opposite terms in $x - 27 \sqrt[3]{4 x} + 27 \sqrt[3]{4 x}$ .

Add $- 27 \sqrt[3]{4 x}$ and $27 \sqrt[3]{4 x}$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (\frac{- 6 - \sqrt[3]{4 x}}{2}) = x$

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

Set up the composite result function.

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

Evaluate $f (- 2 x^{3} - 18 x^{2} - 54 x - 54)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- 6 - \sqrt[3]{4 (- 2 x^{3} - 18 x^{2} - 54 x - 54)}}{2}$

Simplify the numerator.

Factor $- 1$ out of $- 6 - \sqrt[3]{4 (- 2 x^{3} - 18 x^{2} - 54 x - 54)}$ .

Rewrite $- 6$ as $- 1 (6)$ .

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- 1 \cdot 6 - \sqrt[3]{4 (- 2 x^{3} - 18 x^{2} - 54 x - 54)}}{2}$

Factor $- 1$ out of $- \sqrt[3]{4 (- 2 x^{3} - 18 x^{2} - 54 x - 54)}$ .

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- 1 \cdot 6 - (\sqrt[3]{4 (- 2 x^{3} - 18 x^{2} - 54 x - 54)})}{2}$

Factor $- 1$ out of $- 1 (6) - (\sqrt[3]{4 (- 2 x^{3} - 18 x^{2} - 54 x - 54)})$ .

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- 1 (6 + \sqrt[3]{4 (- 2 x^{3} - 18 x^{2} - 54 x - 54)})}{2}$

Rewrite $- 1 (6 + \sqrt[3]{4 (- 2 x^{3} - 18 x^{2} - 54 x - 54)})$ as $- (6 + \sqrt[3]{4 (- 2 x^{3} - 18 x^{2} - 54 x - 54)})$ .

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + \sqrt[3]{4 (- 2 x^{3} - 18 x^{2} - 54 x - 54)})}{2}$

Factor $2$ out of $- 2 x^{3} - 18 x^{2} - 54 x - 54$ .

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

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + \sqrt[3]{4 (2 (- x^{3}) - 18 x^{2} - 54 x - 54)})}{2}$

Factor $2$ out of $- 18 x^{2}$ .

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + \sqrt[3]{4 (2 (- x^{3}) + 2 (- 9 x^{2}) - 54 x - 54)})}{2}$

Factor $2$ out of $- 54 x$ .

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + \sqrt[3]{4 (2 (- x^{3}) + 2 (- 9 x^{2}) + 2 (- 27 x) - 54)})}{2}$

Factor $2$ out of $- 54$ .

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + \sqrt[3]{4 (2 (- x^{3}) + 2 (- 9 x^{2}) + 2 (- 27 x) + 2 (- 27))})}{2}$

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

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + \sqrt[3]{4 (2 (- x^{3} - 9 x^{2}) + 2 (- 27 x) + 2 (- 27))})}{2}$

Factor $2$ out of $2 (- x^{3} - 9 x^{2}) + 2 (- 27 x)$ .

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + \sqrt[3]{4 (2 (- x^{3} - 9 x^{2} - 27 x) + 2 (- 27))})}{2}$

Factor $2$ out of $2 (- x^{3} - 9 x^{2} - 27 x) + 2 (- 27)$ .

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + \sqrt[3]{4 (2 (- x^{3} - 9 x^{2} - 27 x - 27))})}{2}$

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + \sqrt[3]{4 \cdot (2 (- x^{3} - 9 x^{2} - 27 x - 27))})}{2}$

Multiply $4$ by $2$ .

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + \sqrt[3]{8 (- x^{3} - 9 x^{2} - 27 x - 27)})}{2}$

Rewrite $8 (- x^{3} - 9 x^{2} - 27 x - 27)$ as $2^{3} ({(- x)}^{3} - 9 x^{2} - 27 x - 27)$ .

Rewrite $8$ as $2^{3}$ .

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + \sqrt[3]{2^{3} (- x^{3} - 9 x^{2} - 27 x - 27)})}{2}$

Rewrite $- x^{3}$ as ${(- x)}^{3}$ .

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + \sqrt[3]{2^{3} ({(- x)}^{3} - 9 x^{2} - 27 x - 27)})}{2}$

Pull terms out from under the radical.

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + 2 \sqrt[3]{{(- x)}^{3} - 9 x^{2} - 27 x - 27})}{2}$

Apply the product rule to $- x$ .

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + 2 \sqrt[3]{{(- 1)}^{3} x^{3} - 9 x^{2} - 27 x - 27})}{2}$

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

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + 2 \sqrt[3]{- x^{3} - 9 x^{2} - 27 x - 27})}{2}$

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

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

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.

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

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

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

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

Substitute $- 3$ into the polynomial.

$- {(- 3)}^{3} - 9 {(- 3)}^{2} - 27 \cdot - 3 - 27$

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

$- - 27 - 9 {(- 3)}^{2} - 27 \cdot - 3 - 27$

Multiply $- 1$ by $- 27$ .

$27 - 9 {(- 3)}^{2} - 27 \cdot - 3 - 27$

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

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

Multiply $- 9$ by $9$ .

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

Subtract $81$ from $27$ .

$- 54 - 27 \cdot - 3 - 27$

Multiply $- 27$ by $- 3$ .

$- 54 + 81 - 27$

Add $- 54$ and $81$ .

$27 - 27$

Subtract $27$ from $27$ .

$0$

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.

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

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

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

$x$

$3$

$x^{3}$

$9 x^{2}$

$27 x$

$27$

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

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

Multiply the new quotient term by the divisor.

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

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

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

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

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

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

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

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

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

Multiply the new quotient term by the divisor.

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

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

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

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

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

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

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

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

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

Multiply the new quotient term by the divisor.

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

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

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

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

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

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

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

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

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + 2 \sqrt[3]{(x + 3) (- x^{2} - 6 x - 9)})}{2}$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 9 = 9$ and whose sum is $b = - 6$ .

Factor $- 6$ out of $- 6 x$ .

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + 2 \sqrt[3]{(x + 3) (- x^{2} - 6 x - 9)})}{2}$

Rewrite $- 6$ as $- 3$ plus $- 3$

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + 2 \sqrt[3]{(x + 3) (- x^{2} + (- 3 - 3) x - 9)})}{2}$

Apply the distributive property.

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + 2 \sqrt[3]{(x + 3) (- x^{2} - 3 x - 3 x - 9)})}{2}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + 2 \sqrt[3]{(x + 3) ((- x^{2} - 3 x) - 3 x - 9)})}{2}$

Factor out the greatest common factor (GCF) from each group.

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + 2 \sqrt[3]{(x + 3) (x (- x - 3) + 3 (- x - 3))})}{2}$

Factor the polynomial by factoring out the greatest common factor, $- x - 3$ .

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + 2 \sqrt[3]{(x + 3) ((- x - 3) (x + 3))})}{2}$

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + 2 \sqrt[3]{(x + 3) (- x - 3) (x + 3)})}{2}$

Combine exponents.

Factor $- 1$ out of $x$ .

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + 2 \sqrt[3]{(- 1 (- x) + 3) (- x - 3) (x + 3)})}{2}$

Rewrite $3$ as $- 1 (- 3)$ .

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + 2 \sqrt[3]{(- 1 (- x) - 1 \cdot - 3) (- x - 3) (x + 3)})}{2}$

Factor $- 1$ out of $- 1 (- x) - 1 (- 3)$ .

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + 2 \sqrt[3]{- 1 (- x - 3) (- x - 3) (x + 3)})}{2}$

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

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + 2 \sqrt[3]{- 1 ((- x - 3) (- x - 3)) (x + 3)})}{2}$

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

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + 2 \sqrt[3]{- 1 ((- x - 3) (- x - 3)) (x + 3)})}{2}$

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

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + 2 \sqrt[3]{- 1 {(- x - 3)}^{1 + 1} (x + 3)})}{2}$

Add $1$ and $1$ .

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + 2 \sqrt[3]{- 1 {(- x - 3)}^{2} (x + 3)})}{2}$

Factor $- 1$ out of $- x$ .

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + 2 \sqrt[3]{- 1 {(- (x) - 3)}^{2} (x + 3)})}{2}$

Rewrite $- 3$ as $- 1 (3)$ .

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + 2 \sqrt[3]{- 1 {(- (x) - 1 \cdot 3)}^{2} (x + 3)})}{2}$

Factor $- 1$ out of $- (x) - 1 (3)$ .

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + 2 \sqrt[3]{- 1 {(- (x + 3))}^{2} (x + 3)})}{2}$

Rewrite $- (x + 3)$ as $- 1 (x + 3)$ .

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + 2 \sqrt[3]{- 1 {(- 1 (x + 3))}^{2} (x + 3)})}{2}$

Apply the product rule to $- 1 (x + 3)$ .

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + 2 \sqrt[3]{- 1 ({(- 1)}^{2} {(x + 3)}^{2}) (x + 3)})}{2}$

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

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + 2 \sqrt[3]{- 1 (1 {(x + 3)}^{2}) (x + 3)})}{2}$

Multiply ${(x + 3)}^{2}$ by $1$ .

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + 2 \sqrt[3]{- 1 {(x + 3)}^{2} (x + 3)})}{2}$

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

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + 2 \sqrt[3]{- 1 ((x + 3) {(x + 3)}^{2})})}{2}$

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

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + 2 \sqrt[3]{- 1 {(x + 3)}^{1 + 2}})}{2}$

Add $1$ and $2$ .

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + 2 \sqrt[3]{- 1 {(x + 3)}^{3}})}{2}$

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

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + 2 \sqrt[3]{{(- 1 (x + 3))}^{3}})}{2}$

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

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + 2 (- 1 (x + 3)))}{2}$

Apply the distributive property.

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + 2 (- 1 x - 1 \cdot 3))}{2}$

Rewrite $- 1 x$ as $- x$ .

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + 2 (- x - 1 \cdot 3))}{2}$

Multiply $- 1$ by $3$ .

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + 2 (- x - 3))}{2}$

Apply the distributive property.

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 + 2 (- x) + 2 \cdot - 3)}{2}$

Multiply $- 1$ by $2$ .

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 - 2 x + 2 \cdot - 3)}{2}$

Multiply $2$ by $- 3$ .

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (6 - 2 x - 6)}{2}$

Subtract $6$ from $6$ .

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{- (- 2 x + 0)}{2}$

Add $- 2 x$ and $0$ .

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{2 x}{2}$

Multiply $- 1$ by $- 2$ .

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{2 x}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = \frac{2 x}{2}$

Divide $x$ by $1$ .

$f (- 2 x^{3} - 18 x^{2} - 54 x - 54) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - 2 x^{3} - 18 x^{2} - 54 x - 54$ is the inverse of $f (x) = \frac{- 6 - \sqrt[3]{4 x}}{2}$ .

$f^{- 1} (x) = - 2 x^{3} - 18 x^{2} - 54 x - 54$

Find the Inverse f(x)=(-6- cube root of 4x)/2

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