Find the Inverse cube root of x-9

$\sqrt[3]{x} - 9$

Step 1

Interchange the variables.

$x = \sqrt[3]{y} - 9$

Step 2

Solve for $y$ .

Rewrite the equation as $\sqrt[3]{y} - 9 = x$ .

$\sqrt[3]{y} - 9 = x$

Add $9$ to both sides of the equation.

$\sqrt[3]{y} = x + 9$

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

${\sqrt[3]{y}}^{3} = {(x + 9)}^{3}$

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

Simplify ${(x + 9)}^{3}$ .

Use the Binomial Theorem.

$y = x^{3} + 3 x^{2} \cdot 9 + 3 x \cdot 9^{2} + 9^{3}$

Simplify each term.

Multiply $9$ by $3$ .

$y = x^{3} + 27 x^{2} + 3 x \cdot 9^{2} + 9^{3}$

Raise $9$ to the power of $2$ .

$y = x^{3} + 27 x^{2} + 3 x \cdot 81 + 9^{3}$

Multiply $81$ by $3$ .

$y = x^{3} + 27 x^{2} + 243 x + 9^{3}$

Raise $9$ to the power of $3$ .

$y = x^{3} + 27 x^{2} + 243 x + 729$

Step 3

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

$f^{- 1} (x) = x^{3} + 27 x^{2} + 243 x + 729$

Step 4

Verify if $f^{- 1} (x) = x^{3} + 27 x^{2} + 243 x + 729$ is the inverse of $f (x) = \sqrt[3]{x} - 9$ .

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 $f^{- 1} (\sqrt[3]{x} - 9)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\sqrt[3]{x} - 9) = {(\sqrt[3]{x} - 9)}^{3} + 27 {(\sqrt[3]{x} - 9)}^{2} + 243 (\sqrt[3]{x} - 9) + 729$

Simplify each term.

Use the Binomial Theorem.

$f^{- 1} (\sqrt[3]{x} - 9) = {\sqrt[3]{x}}^{3} + 3 {\sqrt[3]{x}}^{2} \cdot - 9 + 3 \sqrt[3]{x} {(- 9)}^{2} + {(- 9)}^{3} + 27 {(\sqrt[3]{x} - 9)}^{2} + 243 (\sqrt[3]{x} - 9) + 729$

Simplify each term.

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

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

$f^{- 1} (\sqrt[3]{x} - 9) = {(x^{\frac{1}{3}})}^{3} + 3 {\sqrt[3]{x}}^{2} \cdot - 9 + 3 \sqrt[3]{x} {(- 9)}^{2} + {(- 9)}^{3} + 27 {(\sqrt[3]{x} - 9)}^{2} + 243 (\sqrt[3]{x} - 9) + 729$

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

$f^{- 1} (\sqrt[3]{x} - 9) = x^{\frac{1}{3} \cdot 3} + 3 {\sqrt[3]{x}}^{2} \cdot - 9 + 3 \sqrt[3]{x} {(- 9)}^{2} + {(- 9)}^{3} + 27 {(\sqrt[3]{x} - 9)}^{2} + 243 (\sqrt[3]{x} - 9) + 729$

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

$f^{- 1} (\sqrt[3]{x} - 9) = x^{\frac{3}{3}} + 3 {\sqrt[3]{x}}^{2} \cdot - 9 + 3 \sqrt[3]{x} {(- 9)}^{2} + {(- 9)}^{3} + 27 {(\sqrt[3]{x} - 9)}^{2} + 243 (\sqrt[3]{x} - 9) + 729$

Cancel the common factor of $3$ .

Cancel the common factor.

$f^{- 1} (\sqrt[3]{x} - 9) = x^{\frac{3}{3}} + 3 {\sqrt[3]{x}}^{2} \cdot - 9 + 3 \sqrt[3]{x} {(- 9)}^{2} + {(- 9)}^{3} + 27 {(\sqrt[3]{x} - 9)}^{2} + 243 (\sqrt[3]{x} - 9) + 729$

Rewrite the expression.

$f^{- 1} (\sqrt[3]{x} - 9) = x + 3 {\sqrt[3]{x}}^{2} \cdot - 9 + 3 \sqrt[3]{x} {(- 9)}^{2} + {(- 9)}^{3} + 27 {(\sqrt[3]{x} - 9)}^{2} + 243 (\sqrt[3]{x} - 9) + 729$

Simplify.

$f^{- 1} (\sqrt[3]{x} - 9) = x + 3 {\sqrt[3]{x}}^{2} \cdot - 9 + 3 \sqrt[3]{x} {(- 9)}^{2} + {(- 9)}^{3} + 27 {(\sqrt[3]{x} - 9)}^{2} + 243 (\sqrt[3]{x} - 9) + 729$

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

$f^{- 1} (\sqrt[3]{x} - 9) = x + 3 \sqrt[3]{x^{2}} \cdot - 9 + 3 \sqrt[3]{x} {(- 9)}^{2} + {(- 9)}^{3} + 27 {(\sqrt[3]{x} - 9)}^{2} + 243 (\sqrt[3]{x} - 9) + 729$

Multiply $- 9$ by $3$ .

$f^{- 1} (\sqrt[3]{x} - 9) = x - 27 \sqrt[3]{x^{2}} + 3 \sqrt[3]{x} {(- 9)}^{2} + {(- 9)}^{3} + 27 {(\sqrt[3]{x} - 9)}^{2} + 243 (\sqrt[3]{x} - 9) + 729$

Raise $- 9$ to the power of $2$ .

$f^{- 1} (\sqrt[3]{x} - 9) = x - 27 \sqrt[3]{x^{2}} + 3 \sqrt[3]{x} \cdot 81 + {(- 9)}^{3} + 27 {(\sqrt[3]{x} - 9)}^{2} + 243 (\sqrt[3]{x} - 9) + 729$

Multiply $81$ by $3$ .

$f^{- 1} (\sqrt[3]{x} - 9) = x - 27 \sqrt[3]{x^{2}} + 243 \sqrt[3]{x} + {(- 9)}^{3} + 27 {(\sqrt[3]{x} - 9)}^{2} + 243 (\sqrt[3]{x} - 9) + 729$

Raise $- 9$ to the power of $3$ .

$f^{- 1} (\sqrt[3]{x} - 9) = x - 27 \sqrt[3]{x^{2}} + 243 \sqrt[3]{x} - 729 + 27 {(\sqrt[3]{x} - 9)}^{2} + 243 (\sqrt[3]{x} - 9) + 729$

Rewrite ${(\sqrt[3]{x} - 9)}^{2}$ as $(\sqrt[3]{x} - 9) (\sqrt[3]{x} - 9)$ .

$f^{- 1} (\sqrt[3]{x} - 9) = x - 27 \sqrt[3]{x^{2}} + 243 \sqrt[3]{x} - 729 + 27 ((\sqrt[3]{x} - 9) (\sqrt[3]{x} - 9)) + 243 (\sqrt[3]{x} - 9) + 729$

Expand $(\sqrt[3]{x} - 9) (\sqrt[3]{x} - 9)$ using the FOIL Method.

Apply the distributive property.

$f^{- 1} (\sqrt[3]{x} - 9) = x - 27 \sqrt[3]{x^{2}} + 243 \sqrt[3]{x} - 729 + 27 (\sqrt[3]{x} (\sqrt[3]{x} - 9) - 9 (\sqrt[3]{x} - 9)) + 243 (\sqrt[3]{x} - 9) + 729$

Apply the distributive property.

$f^{- 1} (\sqrt[3]{x} - 9) = x - 27 \sqrt[3]{x^{2}} + 243 \sqrt[3]{x} - 729 + 27 (\sqrt[3]{x} \sqrt[3]{x} + \sqrt[3]{x} \cdot - 9 - 9 (\sqrt[3]{x} - 9)) + 243 (\sqrt[3]{x} - 9) + 729$

Apply the distributive property.

$f^{- 1} (\sqrt[3]{x} - 9) = x - 27 \sqrt[3]{x^{2}} + 243 \sqrt[3]{x} - 729 + 27 (\sqrt[3]{x} \sqrt[3]{x} + \sqrt[3]{x} \cdot - 9 - 9 \sqrt[3]{x} - 9 \cdot - 9) + 243 (\sqrt[3]{x} - 9) + 729$

Simplify and combine like terms.

Simplify each term.

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

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

$f^{- 1} (\sqrt[3]{x} - 9) = x - 27 \sqrt[3]{x^{2}} + 243 \sqrt[3]{x} - 729 + 27 (\sqrt[3]{x} \sqrt[3]{x} + \sqrt[3]{x} \cdot - 9 - 9 \sqrt[3]{x} - 9 \cdot - 9) + 243 (\sqrt[3]{x} - 9) + 729$

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

$f^{- 1} (\sqrt[3]{x} - 9) = x - 27 \sqrt[3]{x^{2}} + 243 \sqrt[3]{x} - 729 + 27 (\sqrt[3]{x} \sqrt[3]{x} + \sqrt[3]{x} \cdot - 9 - 9 \sqrt[3]{x} - 9 \cdot - 9) + 243 (\sqrt[3]{x} - 9) + 729$

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

$f^{- 1} (\sqrt[3]{x} - 9) = x - 27 \sqrt[3]{x^{2}} + 243 \sqrt[3]{x} - 729 + 27 ({\sqrt[3]{x}}^{1 + 1} + \sqrt[3]{x} \cdot - 9 - 9 \sqrt[3]{x} - 9 \cdot - 9) + 243 (\sqrt[3]{x} - 9) + 729$

Add $1$ and $1$ .

$f^{- 1} (\sqrt[3]{x} - 9) = x - 27 \sqrt[3]{x^{2}} + 243 \sqrt[3]{x} - 729 + 27 ({\sqrt[3]{x}}^{2} + \sqrt[3]{x} \cdot - 9 - 9 \sqrt[3]{x} - 9 \cdot - 9) + 243 (\sqrt[3]{x} - 9) + 729$

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

$f^{- 1} (\sqrt[3]{x} - 9) = x - 27 \sqrt[3]{x^{2}} + 243 \sqrt[3]{x} - 729 + 27 (\sqrt[3]{x^{2}} + \sqrt[3]{x} \cdot - 9 - 9 \sqrt[3]{x} - 9 \cdot - 9) + 243 (\sqrt[3]{x} - 9) + 729$

Move $- 9$ to the left of $\sqrt[3]{x}$ .

$f^{- 1} (\sqrt[3]{x} - 9) = x - 27 \sqrt[3]{x^{2}} + 243 \sqrt[3]{x} - 729 + 27 (\sqrt[3]{x^{2}} - 9 \cdot \sqrt[3]{x} - 9 \sqrt[3]{x} - 9 \cdot - 9) + 243 (\sqrt[3]{x} - 9) + 729$

Multiply $- 9$ by $- 9$ .

$f^{- 1} (\sqrt[3]{x} - 9) = x - 27 \sqrt[3]{x^{2}} + 243 \sqrt[3]{x} - 729 + 27 (\sqrt[3]{x^{2}} - 9 \sqrt[3]{x} - 9 \sqrt[3]{x} + 81) + 243 (\sqrt[3]{x} - 9) + 729$

Subtract $9 \sqrt[3]{x}$ from $- 9 \sqrt[3]{x}$ .

$f^{- 1} (\sqrt[3]{x} - 9) = x - 27 \sqrt[3]{x^{2}} + 243 \sqrt[3]{x} - 729 + 27 (\sqrt[3]{x^{2}} - 18 \sqrt[3]{x} + 81) + 243 (\sqrt[3]{x} - 9) + 729$

Apply the distributive property.

$f^{- 1} (\sqrt[3]{x} - 9) = x - 27 \sqrt[3]{x^{2}} + 243 \sqrt[3]{x} - 729 + 27 \sqrt[3]{x^{2}} + 27 (- 18 \sqrt[3]{x}) + 27 \cdot 81 + 243 (\sqrt[3]{x} - 9) + 729$

Simplify.

Multiply $- 18$ by $27$ .

$f^{- 1} (\sqrt[3]{x} - 9) = x - 27 \sqrt[3]{x^{2}} + 243 \sqrt[3]{x} - 729 + 27 \sqrt[3]{x^{2}} - 486 \sqrt[3]{x} + 27 \cdot 81 + 243 (\sqrt[3]{x} - 9) + 729$

Multiply $27$ by $81$ .

$f^{- 1} (\sqrt[3]{x} - 9) = x - 27 \sqrt[3]{x^{2}} + 243 \sqrt[3]{x} - 729 + 27 \sqrt[3]{x^{2}} - 486 \sqrt[3]{x} + 2187 + 243 (\sqrt[3]{x} - 9) + 729$

Apply the distributive property.

$f^{- 1} (\sqrt[3]{x} - 9) = x - 27 \sqrt[3]{x^{2}} + 243 \sqrt[3]{x} - 729 + 27 \sqrt[3]{x^{2}} - 486 \sqrt[3]{x} + 2187 + 243 \sqrt[3]{x} + 243 \cdot - 9 + 729$

Multiply $243$ by $- 9$ .

$f^{- 1} (\sqrt[3]{x} - 9) = x - 27 \sqrt[3]{x^{2}} + 243 \sqrt[3]{x} - 729 + 27 \sqrt[3]{x^{2}} - 486 \sqrt[3]{x} + 2187 + 243 \sqrt[3]{x} - 2187 + 729$

Simplify by adding terms.

Combine the opposite terms in $x - 27 \sqrt[3]{x^{2}} + 243 \sqrt[3]{x} - 729 + 27 \sqrt[3]{x^{2}} - 486 \sqrt[3]{x} + 2187 + 243 \sqrt[3]{x} - 2187 + 729$ .

Add $- 27 \sqrt[3]{x^{2}}$ and $27 \sqrt[3]{x^{2}}$ .

$f^{- 1} (\sqrt[3]{x} - 9) = x + 0 + 243 \sqrt[3]{x} - 729 - 486 \sqrt[3]{x} + 2187 + 243 \sqrt[3]{x} - 2187 + 729$

Add $x$ and $0$ .

$f^{- 1} (\sqrt[3]{x} - 9) = x + 243 \sqrt[3]{x} - 729 - 486 \sqrt[3]{x} + 2187 + 243 \sqrt[3]{x} - 2187 + 729$

Subtract $2187$ from $2187$ .

$f^{- 1} (\sqrt[3]{x} - 9) = x + 243 \sqrt[3]{x} - 729 - 486 \sqrt[3]{x} + 0 + 243 \sqrt[3]{x} + 729$

Add $x + 243 \sqrt[3]{x} - 729 - 486 \sqrt[3]{x}$ and $0$ .

$f^{- 1} (\sqrt[3]{x} - 9) = x + 243 \sqrt[3]{x} - 729 - 486 \sqrt[3]{x} + 243 \sqrt[3]{x} + 729$

Add $- 729$ and $729$ .

$f^{- 1} (\sqrt[3]{x} - 9) = x + 243 \sqrt[3]{x} + 0 - 486 \sqrt[3]{x} + 243 \sqrt[3]{x}$

Add $x + 243 \sqrt[3]{x}$ and $0$ .

$f^{- 1} (\sqrt[3]{x} - 9) = x + 243 \sqrt[3]{x} - 486 \sqrt[3]{x} + 243 \sqrt[3]{x}$

Subtract $486 \sqrt[3]{x}$ from $243 \sqrt[3]{x}$ .

$f^{- 1} (\sqrt[3]{x} - 9) = x - 243 \sqrt[3]{x} + 243 \sqrt[3]{x}$

Combine the opposite terms in $x - 243 \sqrt[3]{x} + 243 \sqrt[3]{x}$ .

Add $- 243 \sqrt[3]{x}$ and $243 \sqrt[3]{x}$ .

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

Add $x$ and $0$ .

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

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

Set up the composite result function.

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

Evaluate $f (x^{3} + 27 x^{2} + 243 x + 729)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (x^{3} + 27 x^{2} + 243 x + 729) = \sqrt[3]{x^{3} + 27 x^{2} + 243 x + 729} - 9$

Remove parentheses.

$f (x^{3} + 27 x^{2} + 243 x + 729) = \sqrt[3]{x^{3} + 27 x^{2} + 243 x + 729} - 9$

Simplify each term.

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

$f (x^{3} + 27 x^{2} + 243 x + 729) = \sqrt[3]{x^{3} + 3 \cdot (9 x^{2}) + 3 \cdot (9^{2} x) + 9^{3}} - 9$

Factor $x^{3} + 3 \cdot 9 x^{2} + 3 \cdot 9^{2} x + 9^{3}$ using the binomial theorem.

$f (x^{3} + 27 x^{2} + 243 x + 729) = \sqrt[3]{{(x + 9)}^{3}} - 9$

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

$f (x^{3} + 27 x^{2} + 243 x + 729) = x + 9 - 9$

Combine the opposite terms in $x + 9 - 9$ .

Subtract $9$ from $9$ .

$f (x^{3} + 27 x^{2} + 243 x + 729) = x + 0$

Add $x$ and $0$ .

$f (x^{3} + 27 x^{2} + 243 x + 729) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = x^{3} + 27 x^{2} + 243 x + 729$ is the inverse of $f (x) = \sqrt[3]{x} - 9$ .

$f^{- 1} (x) = x^{3} + 27 x^{2} + 243 x + 729$

Find the Inverse cube root of x-9

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