Find the Inverse y = cube root of x-2-4

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

Interchange the variables.

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

Solve for $y$ .

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

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

Add $4$ to both sides of the equation.

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

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

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

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{y - 2}$ as ${(y - 2)}^{\frac{1}{3}}$ .

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

Simplify the left side.

Simplify ${({(y - 2)}^{\frac{1}{3}})}^{3}$ .

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

Simplify ${(x + 4)}^{3}$ .

Use the Binomial Theorem.

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

Simplify each term.

Multiply $4$ by $3$ .

$y - 2 = x^{3} + 12 x^{2} + 3 x \cdot 4^{2} + 4^{3}$

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

$y - 2 = x^{3} + 12 x^{2} + 3 x \cdot 16 + 4^{3}$

Multiply $16$ by $3$ .

$y - 2 = x^{3} + 12 x^{2} + 48 x + 4^{3}$

Raise $4$ to the power of $3$ .

$y - 2 = x^{3} + 12 x^{2} + 48 x + 64$

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

Add $2$ to both sides of the equation.

$y = x^{3} + 12 x^{2} + 48 x + 64 + 2$

Add $64$ and $2$ .

$y = x^{3} + 12 x^{2} + 48 x + 66$

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

$f^{- 1} (x) = x^{3} + 12 x^{2} + 48 x + 66$

Verify if $f^{- 1} (x) = x^{3} + 12 x^{2} + 48 x + 66$ is the inverse of $f (x) = \sqrt[3]{x - 2} - 4$ .

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

$f^{- 1} (\sqrt[3]{x - 2} - 4) = {(\sqrt[3]{x - 2} - 4)}^{3} + 12 {(\sqrt[3]{x - 2} - 4)}^{2} + 48 (\sqrt[3]{x - 2} - 4) + 66$

Simplify each term.

Use the Binomial Theorem.

$f^{- 1} (\sqrt[3]{x - 2} - 4) = {\sqrt[3]{x - 2}}^{3} + 3 {\sqrt[3]{x - 2}}^{2} \cdot - 4 + 3 \sqrt[3]{x - 2} {(- 4)}^{2} + {(- 4)}^{3} + 12 {(\sqrt[3]{x - 2} - 4)}^{2} + 48 (\sqrt[3]{x - 2} - 4) + 66$

Simplify each term.

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x - 2}$ as ${(x - 2)}^{\frac{1}{3}}$ .

$f^{- 1} (\sqrt[3]{x - 2} - 4) = {({(x - 2)}^{\frac{1}{3}})}^{3} + 3 {\sqrt[3]{x - 2}}^{2} \cdot - 4 + 3 \sqrt[3]{x - 2} {(- 4)}^{2} + {(- 4)}^{3} + 12 {(\sqrt[3]{x - 2} - 4)}^{2} + 48 (\sqrt[3]{x - 2} - 4) + 66$

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

$f^{- 1} (\sqrt[3]{x - 2} - 4) = {(x - 2)}^{\frac{1}{3} \cdot 3} + 3 {\sqrt[3]{x - 2}}^{2} \cdot - 4 + 3 \sqrt[3]{x - 2} {(- 4)}^{2} + {(- 4)}^{3} + 12 {(\sqrt[3]{x - 2} - 4)}^{2} + 48 (\sqrt[3]{x - 2} - 4) + 66$

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

$f^{- 1} (\sqrt[3]{x - 2} - 4) = {(x - 2)}^{\frac{3}{3}} + 3 {\sqrt[3]{x - 2}}^{2} \cdot - 4 + 3 \sqrt[3]{x - 2} {(- 4)}^{2} + {(- 4)}^{3} + 12 {(\sqrt[3]{x - 2} - 4)}^{2} + 48 (\sqrt[3]{x - 2} - 4) + 66$

Cancel the common factor of $3$ .

Cancel the common factor.

Rewrite the expression.

$f^{- 1} (\sqrt[3]{x - 2} - 4) = (x - 2) + 3 {\sqrt[3]{x - 2}}^{2} \cdot - 4 + 3 \sqrt[3]{x - 2} {(- 4)}^{2} + {(- 4)}^{3} + 12 {(\sqrt[3]{x - 2} - 4)}^{2} + 48 (\sqrt[3]{x - 2} - 4) + 66$

Simplify.

$f^{- 1} (\sqrt[3]{x - 2} - 4) = x - 2 + 3 {\sqrt[3]{x - 2}}^{2} \cdot - 4 + 3 \sqrt[3]{x - 2} {(- 4)}^{2} + {(- 4)}^{3} + 12 {(\sqrt[3]{x - 2} - 4)}^{2} + 48 (\sqrt[3]{x - 2} - 4) + 66$

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

$f^{- 1} (\sqrt[3]{x - 2} - 4) = x - 2 + 3 \sqrt[3]{{(x - 2)}^{2}} \cdot - 4 + 3 \sqrt[3]{x - 2} {(- 4)}^{2} + {(- 4)}^{3} + 12 {(\sqrt[3]{x - 2} - 4)}^{2} + 48 (\sqrt[3]{x - 2} - 4) + 66$

Multiply $- 4$ by $3$ .

$f^{- 1} (\sqrt[3]{x - 2} - 4) = x - 2 - 12 \sqrt[3]{{(x - 2)}^{2}} + 3 \sqrt[3]{x - 2} {(- 4)}^{2} + {(- 4)}^{3} + 12 {(\sqrt[3]{x - 2} - 4)}^{2} + 48 (\sqrt[3]{x - 2} - 4) + 66$

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

$f^{- 1} (\sqrt[3]{x - 2} - 4) = x - 2 - 12 \sqrt[3]{{(x - 2)}^{2}} + 3 \sqrt[3]{x - 2} \cdot 16 + {(- 4)}^{3} + 12 {(\sqrt[3]{x - 2} - 4)}^{2} + 48 (\sqrt[3]{x - 2} - 4) + 66$

Multiply $16$ by $3$ .

$f^{- 1} (\sqrt[3]{x - 2} - 4) = x - 2 - 12 \sqrt[3]{{(x - 2)}^{2}} + 48 \sqrt[3]{x - 2} + {(- 4)}^{3} + 12 {(\sqrt[3]{x - 2} - 4)}^{2} + 48 (\sqrt[3]{x - 2} - 4) + 66$

Raise $- 4$ to the power of $3$ .

$f^{- 1} (\sqrt[3]{x - 2} - 4) = x - 2 - 12 \sqrt[3]{{(x - 2)}^{2}} + 48 \sqrt[3]{x - 2} - 64 + 12 {(\sqrt[3]{x - 2} - 4)}^{2} + 48 (\sqrt[3]{x - 2} - 4) + 66$

Subtract $64$ from $- 2$ .

$f^{- 1} (\sqrt[3]{x - 2} - 4) = x - 66 - 12 \sqrt[3]{{(x - 2)}^{2}} + 48 \sqrt[3]{x - 2} + 12 {(\sqrt[3]{x - 2} - 4)}^{2} + 48 (\sqrt[3]{x - 2} - 4) + 66$

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

$f^{- 1} (\sqrt[3]{x - 2} - 4) = x - 66 - 12 \sqrt[3]{{(x - 2)}^{2}} + 48 \sqrt[3]{x - 2} + 12 ((\sqrt[3]{x - 2} - 4) (\sqrt[3]{x - 2} - 4)) + 48 (\sqrt[3]{x - 2} - 4) + 66$

Expand $(\sqrt[3]{x - 2} - 4) (\sqrt[3]{x - 2} - 4)$ using the FOIL Method.

Apply the distributive property.

$f^{- 1} (\sqrt[3]{x - 2} - 4) = x - 66 - 12 \sqrt[3]{{(x - 2)}^{2}} + 48 \sqrt[3]{x - 2} + 12 (\sqrt[3]{x - 2} (\sqrt[3]{x - 2} - 4) - 4 (\sqrt[3]{x - 2} - 4)) + 48 (\sqrt[3]{x - 2} - 4) + 66$

Apply the distributive property.

$f^{- 1} (\sqrt[3]{x - 2} - 4) = x - 66 - 12 \sqrt[3]{{(x - 2)}^{2}} + 48 \sqrt[3]{x - 2} + 12 (\sqrt[3]{x - 2} \sqrt[3]{x - 2} + \sqrt[3]{x - 2} \cdot - 4 - 4 (\sqrt[3]{x - 2} - 4)) + 48 (\sqrt[3]{x - 2} - 4) + 66$

Apply the distributive property.

$f^{- 1} (\sqrt[3]{x - 2} - 4) = x - 66 - 12 \sqrt[3]{{(x - 2)}^{2}} + 48 \sqrt[3]{x - 2} + 12 (\sqrt[3]{x - 2} \sqrt[3]{x - 2} + \sqrt[3]{x - 2} \cdot - 4 - 4 \sqrt[3]{x - 2} - 4 \cdot - 4) + 48 (\sqrt[3]{x - 2} - 4) + 66$

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt[3]{x - 2} \sqrt[3]{x - 2}$ .

Raise $\sqrt[3]{x - 2}$ to the power of $1$ .

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

$f^{- 1} (\sqrt[3]{x - 2} - 4) = x - 66 - 12 \sqrt[3]{{(x - 2)}^{2}} + 48 \sqrt[3]{x - 2} + 12 ({\sqrt[3]{x - 2}}^{1 + 1} + \sqrt[3]{x - 2} \cdot - 4 - 4 \sqrt[3]{x - 2} - 4 \cdot - 4) + 48 (\sqrt[3]{x - 2} - 4) + 66$

Add $1$ and $1$ .

$f^{- 1} (\sqrt[3]{x - 2} - 4) = x - 66 - 12 \sqrt[3]{{(x - 2)}^{2}} + 48 \sqrt[3]{x - 2} + 12 ({\sqrt[3]{x - 2}}^{2} + \sqrt[3]{x - 2} \cdot - 4 - 4 \sqrt[3]{x - 2} - 4 \cdot - 4) + 48 (\sqrt[3]{x - 2} - 4) + 66$

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

$f^{- 1} (\sqrt[3]{x - 2} - 4) = x - 66 - 12 \sqrt[3]{{(x - 2)}^{2}} + 48 \sqrt[3]{x - 2} + 12 (\sqrt[3]{{(x - 2)}^{2}} + \sqrt[3]{x - 2} \cdot - 4 - 4 \sqrt[3]{x - 2} - 4 \cdot - 4) + 48 (\sqrt[3]{x - 2} - 4) + 66$

Move $- 4$ to the left of $\sqrt[3]{x - 2}$ .

$f^{- 1} (\sqrt[3]{x - 2} - 4) = x - 66 - 12 \sqrt[3]{{(x - 2)}^{2}} + 48 \sqrt[3]{x - 2} + 12 (\sqrt[3]{{(x - 2)}^{2}} - 4 \cdot \sqrt[3]{x - 2} - 4 \sqrt[3]{x - 2} - 4 \cdot - 4) + 48 (\sqrt[3]{x - 2} - 4) + 66$

Multiply $- 4$ by $- 4$ .

$f^{- 1} (\sqrt[3]{x - 2} - 4) = x - 66 - 12 \sqrt[3]{{(x - 2)}^{2}} + 48 \sqrt[3]{x - 2} + 12 (\sqrt[3]{{(x - 2)}^{2}} - 4 \sqrt[3]{x - 2} - 4 \sqrt[3]{x - 2} + 16) + 48 (\sqrt[3]{x - 2} - 4) + 66$

Subtract $4 \sqrt[3]{x - 2}$ from $- 4 \sqrt[3]{x - 2}$ .

$f^{- 1} (\sqrt[3]{x - 2} - 4) = x - 66 - 12 \sqrt[3]{{(x - 2)}^{2}} + 48 \sqrt[3]{x - 2} + 12 (\sqrt[3]{{(x - 2)}^{2}} - 8 \sqrt[3]{x - 2} + 16) + 48 (\sqrt[3]{x - 2} - 4) + 66$

Apply the distributive property.

$f^{- 1} (\sqrt[3]{x - 2} - 4) = x - 66 - 12 \sqrt[3]{{(x - 2)}^{2}} + 48 \sqrt[3]{x - 2} + 12 \sqrt[3]{{(x - 2)}^{2}} + 12 (- 8 \sqrt[3]{x - 2}) + 12 \cdot 16 + 48 (\sqrt[3]{x - 2} - 4) + 66$

Simplify.

Multiply $- 8$ by $12$ .

$f^{- 1} (\sqrt[3]{x - 2} - 4) = x - 66 - 12 \sqrt[3]{{(x - 2)}^{2}} + 48 \sqrt[3]{x - 2} + 12 \sqrt[3]{{(x - 2)}^{2}} - 96 \sqrt[3]{x - 2} + 12 \cdot 16 + 48 (\sqrt[3]{x - 2} - 4) + 66$

Multiply $12$ by $16$ .

$f^{- 1} (\sqrt[3]{x - 2} - 4) = x - 66 - 12 \sqrt[3]{{(x - 2)}^{2}} + 48 \sqrt[3]{x - 2} + 12 \sqrt[3]{{(x - 2)}^{2}} - 96 \sqrt[3]{x - 2} + 192 + 48 (\sqrt[3]{x - 2} - 4) + 66$

Apply the distributive property.

$f^{- 1} (\sqrt[3]{x - 2} - 4) = x - 66 - 12 \sqrt[3]{{(x - 2)}^{2}} + 48 \sqrt[3]{x - 2} + 12 \sqrt[3]{{(x - 2)}^{2}} - 96 \sqrt[3]{x - 2} + 192 + 48 \sqrt[3]{x - 2} + 48 \cdot - 4 + 66$

Multiply $48$ by $- 4$ .

$f^{- 1} (\sqrt[3]{x - 2} - 4) = x - 66 - 12 \sqrt[3]{{(x - 2)}^{2}} + 48 \sqrt[3]{x - 2} + 12 \sqrt[3]{{(x - 2)}^{2}} - 96 \sqrt[3]{x - 2} + 192 + 48 \sqrt[3]{x - 2} - 192 + 66$

Simplify by adding terms.

Combine the opposite terms in $x - 66 - 12 \sqrt[3]{{(x - 2)}^{2}} + 48 \sqrt[3]{x - 2} + 12 \sqrt[3]{{(x - 2)}^{2}} - 96 \sqrt[3]{x - 2} + 192 + 48 \sqrt[3]{x - 2} - 192 + 66$ .

Add $- 12 \sqrt[3]{{(x - 2)}^{2}}$ and $12 \sqrt[3]{{(x - 2)}^{2}}$ .

$f^{- 1} (\sqrt[3]{x - 2} - 4) = x - 66 + 0 + 48 \sqrt[3]{x - 2} - 96 \sqrt[3]{x - 2} + 192 + 48 \sqrt[3]{x - 2} - 192 + 66$

Add $x - 66$ and $0$ .

$f^{- 1} (\sqrt[3]{x - 2} - 4) = x - 66 + 48 \sqrt[3]{x - 2} - 96 \sqrt[3]{x - 2} + 192 + 48 \sqrt[3]{x - 2} - 192 + 66$

Subtract $192$ from $192$ .

$f^{- 1} (\sqrt[3]{x - 2} - 4) = x - 66 + 48 \sqrt[3]{x - 2} - 96 \sqrt[3]{x - 2} + 0 + 48 \sqrt[3]{x - 2} + 66$

Add $x - 66 + 48 \sqrt[3]{x - 2} - 96 \sqrt[3]{x - 2}$ and $0$ .

$f^{- 1} (\sqrt[3]{x - 2} - 4) = x - 66 + 48 \sqrt[3]{x - 2} - 96 \sqrt[3]{x - 2} + 48 \sqrt[3]{x - 2} + 66$

Add $- 66$ and $66$ .

$f^{- 1} (\sqrt[3]{x - 2} - 4) = x + 0 + 48 \sqrt[3]{x - 2} - 96 \sqrt[3]{x - 2} + 48 \sqrt[3]{x - 2}$

Add $x$ and $0$ .

$f^{- 1} (\sqrt[3]{x - 2} - 4) = x + 48 \sqrt[3]{x - 2} - 96 \sqrt[3]{x - 2} + 48 \sqrt[3]{x - 2}$

Subtract $96 \sqrt[3]{x - 2}$ from $48 \sqrt[3]{x - 2}$ .

$f^{- 1} (\sqrt[3]{x - 2} - 4) = x - 48 \sqrt[3]{x - 2} + 48 \sqrt[3]{x - 2}$

Combine the opposite terms in $x - 48 \sqrt[3]{x - 2} + 48 \sqrt[3]{x - 2}$ .

Add $- 48 \sqrt[3]{x - 2}$ and $48 \sqrt[3]{x - 2}$ .

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

Add $x$ and $0$ .

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

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

Set up the composite result function.

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

Evaluate $f (x^{3} + 12 x^{2} + 48 x + 66)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (x^{3} + 12 x^{2} + 48 x + 66) = \sqrt[3]{(x^{3} + 12 x^{2} + 48 x + 66) - 2} - 4$

Simplify each term.

Subtract $2$ from $66$ .

$f (x^{3} + 12 x^{2} + 48 x + 66) = \sqrt[3]{x^{3} + 12 x^{2} + 48 x + 64} - 4$

Rewrite $x^{3} + 12 x^{2} + 48 x + 64$ in a factored form.

Regroup terms.

$f (x^{3} + 12 x^{2} + 48 x + 66) = \sqrt[3]{x^{3} + 64 + 12 x^{2} + 48 x} - 4$

Rewrite $64$ as $4^{3}$ .

$f (x^{3} + 12 x^{2} + 48 x + 66) = \sqrt[3]{x^{3} + 4^{3} + 12 x^{2} + 48 x} - 4$

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = 4$ .

$f (x^{3} + 12 x^{2} + 48 x + 66) = \sqrt[3]{(x + 4) (x^{2} - x \cdot 4 + 4^{2}) + 12 x^{2} + 48 x} - 4$

Simplify.

Multiply $4$ by $- 1$ .

$f (x^{3} + 12 x^{2} + 48 x + 66) = \sqrt[3]{(x + 4) (x^{2} - 4 x + 4^{2}) + 12 x^{2} + 48 x} - 4$

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

$f (x^{3} + 12 x^{2} + 48 x + 66) = \sqrt[3]{(x + 4) (x^{2} - 4 x + 16) + 12 x^{2} + 48 x} - 4$

Factor $12 x$ out of $12 x^{2} + 48 x$ .

Factor $12 x$ out of $12 x^{2}$ .

$f (x^{3} + 12 x^{2} + 48 x + 66) = \sqrt[3]{(x + 4) (x^{2} - 4 x + 16) + 12 x (x) + 48 x} - 4$

Factor $12 x$ out of $48 x$ .

$f (x^{3} + 12 x^{2} + 48 x + 66) = \sqrt[3]{(x + 4) (x^{2} - 4 x + 16) + 12 x (x) + 12 x (4)} - 4$

Factor $12 x$ out of $12 x (x) + 12 x (4)$ .

$f (x^{3} + 12 x^{2} + 48 x + 66) = \sqrt[3]{(x + 4) (x^{2} - 4 x + 16) + 12 x (x + 4)} - 4$

Factor $x + 4$ out of $(x + 4) (x^{2} - 4 x + 16) + 12 x (x + 4)$ .

Factor $x + 4$ out of $12 x (x + 4)$ .

$f (x^{3} + 12 x^{2} + 48 x + 66) = \sqrt[3]{(x + 4) (x^{2} - 4 x + 16) + (x + 4) (12 x)} - 4$

Factor $x + 4$ out of $(x + 4) (x^{2} - 4 x + 16) + (x + 4) (12 x)$ .

$f (x^{3} + 12 x^{2} + 48 x + 66) = \sqrt[3]{(x + 4) (x^{2} - 4 x + 16 + 12 x)} - 4$

Add $- 4 x$ and $12 x$ .

$f (x^{3} + 12 x^{2} + 48 x + 66) = \sqrt[3]{(x + 4) (x^{2} + 8 x + 16)} - 4$

Factor using the perfect square rule.

Rewrite $16$ as $4^{2}$ .

$f (x^{3} + 12 x^{2} + 48 x + 66) = \sqrt[3]{(x + 4) (x^{2} + 8 x + 4^{2})} - 4$

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

$8 x = 2 \cdot x \cdot 4$

Rewrite the polynomial.

$f (x^{3} + 12 x^{2} + 48 x + 66) = \sqrt[3]{(x + 4) (x^{2} + 2 \cdot x \cdot 4 + 4^{2})} - 4$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 4$ .

$f (x^{3} + 12 x^{2} + 48 x + 66) = \sqrt[3]{(x + 4) {(x + 4)}^{2}} - 4$

Multiply $x + 4$ by ${(x + 4)}^{2}$ by adding the exponents.

Multiply $x + 4$ by ${(x + 4)}^{2}$ .

Raise $x + 4$ to the power of $1$ .

$f (x^{3} + 12 x^{2} + 48 x + 66) = \sqrt[3]{(x + 4) {(x + 4)}^{2}} - 4$

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

$f (x^{3} + 12 x^{2} + 48 x + 66) = \sqrt[3]{{(x + 4)}^{1 + 2}} - 4$

Add $1$ and $2$ .

$f (x^{3} + 12 x^{2} + 48 x + 66) = \sqrt[3]{{(x + 4)}^{3}} - 4$

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

$f (x^{3} + 12 x^{2} + 48 x + 66) = x + 4 - 4$

Combine the opposite terms in $x + 4 - 4$ .

Subtract $4$ from $4$ .

$f (x^{3} + 12 x^{2} + 48 x + 66) = x + 0$

Add $x$ and $0$ .

$f (x^{3} + 12 x^{2} + 48 x + 66) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = x^{3} + 12 x^{2} + 48 x + 66$ is the inverse of $f (x) = \sqrt[3]{x - 2} - 4$ .

$f^{- 1} (x) = x^{3} + 12 x^{2} + 48 x + 66$

Find the Inverse y = cube root of x-2-4

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