Find the Inverse 9+ cube root of x

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

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

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

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

Subtract $9$ from 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.

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

Simplify each term.

Multiply $- 9$ by $3$ .

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

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 $f (x) = 9 + \sqrt[3]{x}$ .

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

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

Simplify each term.

Use the Binomial Theorem.

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

Simplify each term.

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

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

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

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

Multiply $3$ by $81$ .

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

Multiply $3$ by $9$ .

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

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

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

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

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

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $9$ by $9$ .

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

Apply the distributive property.

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

Simplify.

Multiply $- 27$ by $81$ .

$f^{- 1} (9 + \sqrt[3]{x}) = 729 + 243 \sqrt[3]{x} + 27 \sqrt[3]{x^{2}} + x - 2187 - 27 (18 \sqrt[3]{x}) - 27 \sqrt[3]{x^{2}} + 243 (9 + \sqrt[3]{x}) - 729$

Multiply $18$ by $- 27$ .

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

Apply the distributive property.

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

Multiply $243$ by $9$ .

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

Simplify by adding terms.

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

Subtract $27 \sqrt[3]{x^{2}}$ from $27 \sqrt[3]{x^{2}}$ .

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

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

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

Add $- 2187$ and $2187$ .

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

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

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

Subtract $729$ from $729$ .

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

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

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

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

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

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

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

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

Add $x$ and $0$ .

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

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 $f$ .

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

Simplify each term.

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

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

Factor $x^{3} - 3 \cdot 9 x^{2} + 3 \cdot 9^{2} x - 1 \cdot 9^{3}$ using the binomial theorem.

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

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

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

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

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 $f (x) = 9 + \sqrt[3]{x}$ .

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

Find the Inverse 9+ cube root of x

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