Find the Inverse f(x)=(x^(1/3)+10)/3

$f (x) = \frac{x^{\frac{1}{3}} + 10}{3}$

Step 1

Write $f (x) = \frac{x^{\frac{1}{3}} + 10}{3}$ as an equation.

$y = \frac{x^{\frac{1}{3}} + 10}{3}$

Step 2

Interchange the variables.

$x = \frac{y^{\frac{1}{3}} + 10}{3}$

Step 3

Solve for $y$ .

Rewrite the equation as $\frac{y^{\frac{1}{3}} + 10}{3} = x$ .

$\frac{y^{\frac{1}{3}} + 10}{3} = x$

Multiply both sides of the equation by $3$ .

$3 \frac{y^{\frac{1}{3}} + 10}{3} = 3 x$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$3 \frac{y^{\frac{1}{3}} + 10}{3} = 3 x$

Rewrite the expression.

$y^{\frac{1}{3}} + 10 = 3 x$

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

Subtract $10$ from both sides of the equation.

$y^{\frac{1}{3}} - 3 x = - 10$

Add $3 x$ to both sides of the equation.

$y^{\frac{1}{3}} = - 10 + 3 x$

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${(y^{\frac{1}{3}})}^{3} = {(- 10 + 3 x)}^{3}$

Simplify the exponent.

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, ${(a^{m})}^{n} = a^{m n}$ .

$y^{\frac{1}{3} \cdot 3} = {(- 10 + 3 x)}^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$y^{\frac{1}{3} \cdot 3} = {(- 10 + 3 x)}^{3}$

Rewrite the expression.

$y^{1} = {(- 10 + 3 x)}^{3}$

Simplify.

$y = {(- 10 + 3 x)}^{3}$

Simplify the right side.

Simplify ${(- 10 + 3 x)}^{3}$ .

Use the Binomial Theorem.

$y = {(- 10)}^{3} + 3 {(- 10)}^{2} (3 x) + 3 \cdot - 10 {(3 x)}^{2} + {(3 x)}^{3}$

Simplify each term.

Raise $- 10$ to the power of $3$ .

$y = - 1000 + 3 {(- 10)}^{2} (3 x) + 3 \cdot - 10 {(3 x)}^{2} + {(3 x)}^{3}$

Raise $- 10$ to the power of $2$ .

$y = - 1000 + 3 \cdot 100 (3 x) + 3 \cdot - 10 {(3 x)}^{2} + {(3 x)}^{3}$

Multiply $3$ by $100$ .

$y = - 1000 + 300 (3 x) + 3 \cdot - 10 {(3 x)}^{2} + {(3 x)}^{3}$

Multiply $3$ by $300$ .

$y = - 1000 + 900 x + 3 \cdot - 10 {(3 x)}^{2} + {(3 x)}^{3}$

Multiply $3$ by $- 10$ .

$y = - 1000 + 900 x - 30 {(3 x)}^{2} + {(3 x)}^{3}$

Apply the product rule to $3 x$ .

$y = - 1000 + 900 x - 30 (3^{2} x^{2}) + {(3 x)}^{3}$

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

$y = - 1000 + 900 x - 30 (9 x^{2}) + {(3 x)}^{3}$

Multiply $9$ by $- 30$ .

$y = - 1000 + 900 x - 270 x^{2} + {(3 x)}^{3}$

Apply the product rule to $3 x$ .

$y = - 1000 + 900 x - 270 x^{2} + 3^{3} x^{3}$

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

$y = - 1000 + 900 x - 270 x^{2} + 27 x^{3}$

Simplify $- 1000 + 900 x - 270 x^{2} + 27 x^{3}$ .

Move $- 1000$ .

$y = 900 x - 270 x^{2} + 27 x^{3} - 1000$

Move $900 x$ .

$y = - 270 x^{2} + 27 x^{3} + 900 x - 1000$

Reorder $- 270 x^{2}$ and $27 x^{3}$ .

$y = 27 x^{3} - 270 x^{2} + 900 x - 1000$

Step 4

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

$f^{- 1} (x) = 27 x^{3} - 270 x^{2} + 900 x - 1000$

Step 5

Verify if $f^{- 1} (x) = 27 x^{3} - 270 x^{2} + 900 x - 1000$ is the inverse of $f (x) = \frac{x^{\frac{1}{3}} + 10}{3}$ .

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 $f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = 27 {(\frac{x^{\frac{1}{3}} + 10}{3})}^{3} - 270 {(\frac{x^{\frac{1}{3}} + 10}{3})}^{2} + 900 (\frac{x^{\frac{1}{3}} + 10}{3}) - 1000$

Simplify each term.

Apply the product rule to $\frac{x^{\frac{1}{3}} + 10}{3}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = 27 (\frac{{(x^{\frac{1}{3}} + 10)}^{3}}{3^{3}}) - 270 {(\frac{x^{\frac{1}{3}} + 10}{3})}^{2} + 900 (\frac{x^{\frac{1}{3}} + 10}{3}) - 1000$

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

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = 27 (\frac{{(x^{\frac{1}{3}} + 10)}^{3}}{27}) - 270 {(\frac{x^{\frac{1}{3}} + 10}{3})}^{2} + 900 (\frac{x^{\frac{1}{3}} + 10}{3}) - 1000$

Cancel the common factor of $27$ .

Cancel the common factor.

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = 27 (\frac{{(x^{\frac{1}{3}} + 10)}^{3}}{27}) - 270 {(\frac{x^{\frac{1}{3}} + 10}{3})}^{2} + 900 (\frac{x^{\frac{1}{3}} + 10}{3}) - 1000$

Rewrite the expression.

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = {(x^{\frac{1}{3}} + 10)}^{3} - 270 {(\frac{x^{\frac{1}{3}} + 10}{3})}^{2} + 900 (\frac{x^{\frac{1}{3}} + 10}{3}) - 1000$

Use the Binomial Theorem.

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = {(x^{\frac{1}{3}})}^{3} + 3 {(x^{\frac{1}{3}})}^{2} \cdot 10 + 3 x^{\frac{1}{3}} \cdot 10^{2} + 10^{3} - 270 {(\frac{x^{\frac{1}{3}} + 10}{3})}^{2} + 900 (\frac{x^{\frac{1}{3}} + 10}{3}) - 1000$

Simplify each term.

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

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

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x^{\frac{1}{3} \cdot 3} + 3 {(x^{\frac{1}{3}})}^{2} \cdot 10 + 3 x^{\frac{1}{3}} \cdot 10^{2} + 10^{3} - 270 {(\frac{x^{\frac{1}{3}} + 10}{3})}^{2} + 900 (\frac{x^{\frac{1}{3}} + 10}{3}) - 1000$

Cancel the common factor of $3$ .

Cancel the common factor.

Rewrite the expression.

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 3 {(x^{\frac{1}{3}})}^{2} \cdot 10 + 3 x^{\frac{1}{3}} \cdot 10^{2} + 10^{3} - 270 {(\frac{x^{\frac{1}{3}} + 10}{3})}^{2} + 900 (\frac{x^{\frac{1}{3}} + 10}{3}) - 1000$

Simplify.

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

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

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 3 x^{\frac{1}{3} \cdot 2} \cdot 10 + 3 x^{\frac{1}{3}} \cdot 10^{2} + 10^{3} - 270 {(\frac{x^{\frac{1}{3}} + 10}{3})}^{2} + 900 (\frac{x^{\frac{1}{3}} + 10}{3}) - 1000$

Combine $\frac{1}{3}$ and $2$ .

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 3 x^{\frac{2}{3}} \cdot 10 + 3 x^{\frac{1}{3}} \cdot 10^{2} + 10^{3} - 270 {(\frac{x^{\frac{1}{3}} + 10}{3})}^{2} + 900 (\frac{x^{\frac{1}{3}} + 10}{3}) - 1000$

Multiply $10$ by $3$ .

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 30 x^{\frac{2}{3}} + 3 x^{\frac{1}{3}} \cdot 10^{2} + 10^{3} - 270 {(\frac{x^{\frac{1}{3}} + 10}{3})}^{2} + 900 (\frac{x^{\frac{1}{3}} + 10}{3}) - 1000$

Raise $10$ to the power of $2$ .

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 30 x^{\frac{2}{3}} + 3 x^{\frac{1}{3}} \cdot 100 + 10^{3} - 270 {(\frac{x^{\frac{1}{3}} + 10}{3})}^{2} + 900 (\frac{x^{\frac{1}{3}} + 10}{3}) - 1000$

Multiply $100$ by $3$ .

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 30 x^{\frac{2}{3}} + 300 x^{\frac{1}{3}} + 10^{3} - 270 {(\frac{x^{\frac{1}{3}} + 10}{3})}^{2} + 900 (\frac{x^{\frac{1}{3}} + 10}{3}) - 1000$

Raise $10$ to the power of $3$ .

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 30 x^{\frac{2}{3}} + 300 x^{\frac{1}{3}} + 1000 - 270 {(\frac{x^{\frac{1}{3}} + 10}{3})}^{2} + 900 (\frac{x^{\frac{1}{3}} + 10}{3}) - 1000$

Apply the product rule to $\frac{x^{\frac{1}{3}} + 10}{3}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 30 x^{\frac{2}{3}} + 300 x^{\frac{1}{3}} + 1000 - 270 \frac{{(x^{\frac{1}{3}} + 10)}^{2}}{3^{2}} + 900 (\frac{x^{\frac{1}{3}} + 10}{3}) - 1000$

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

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 30 x^{\frac{2}{3}} + 300 x^{\frac{1}{3}} + 1000 - 270 \frac{{(x^{\frac{1}{3}} + 10)}^{2}}{9} + 900 (\frac{x^{\frac{1}{3}} + 10}{3}) - 1000$

Cancel the common factor of $9$ .

Factor $9$ out of $- 270$ .

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 30 x^{\frac{2}{3}} + 300 x^{\frac{1}{3}} + 1000 + 9 (- 30) (\frac{{(x^{\frac{1}{3}} + 10)}^{2}}{9}) + 900 (\frac{x^{\frac{1}{3}} + 10}{3}) - 1000$

Cancel the common factor.

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 30 x^{\frac{2}{3}} + 300 x^{\frac{1}{3}} + 1000 + 9 \cdot (- 30 \frac{{(x^{\frac{1}{3}} + 10)}^{2}}{9}) + 900 (\frac{x^{\frac{1}{3}} + 10}{3}) - 1000$

Rewrite the expression.

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 30 x^{\frac{2}{3}} + 300 x^{\frac{1}{3}} + 1000 - 30 {(x^{\frac{1}{3}} + 10)}^{2} + 900 (\frac{x^{\frac{1}{3}} + 10}{3}) - 1000$

Rewrite ${(x^{\frac{1}{3}} + 10)}^{2}$ as $(x^{\frac{1}{3}} + 10) (x^{\frac{1}{3}} + 10)$ .

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 30 x^{\frac{2}{3}} + 300 x^{\frac{1}{3}} + 1000 - 30 ((x^{\frac{1}{3}} + 10) (x^{\frac{1}{3}} + 10)) + 900 (\frac{x^{\frac{1}{3}} + 10}{3}) - 1000$

Expand $(x^{\frac{1}{3}} + 10) (x^{\frac{1}{3}} + 10)$ using the FOIL Method.

Apply the distributive property.

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 30 x^{\frac{2}{3}} + 300 x^{\frac{1}{3}} + 1000 - 30 (x^{\frac{1}{3}} (x^{\frac{1}{3}} + 10) + 10 (x^{\frac{1}{3}} + 10)) + 900 (\frac{x^{\frac{1}{3}} + 10}{3}) - 1000$

Apply the distributive property.

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 30 x^{\frac{2}{3}} + 300 x^{\frac{1}{3}} + 1000 - 30 (x^{\frac{1}{3}} x^{\frac{1}{3}} + x^{\frac{1}{3}} \cdot 10 + 10 (x^{\frac{1}{3}} + 10)) + 900 (\frac{x^{\frac{1}{3}} + 10}{3}) - 1000$

Apply the distributive property.

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 30 x^{\frac{2}{3}} + 300 x^{\frac{1}{3}} + 1000 - 30 (x^{\frac{1}{3}} x^{\frac{1}{3}} + x^{\frac{1}{3}} \cdot 10 + 10 x^{\frac{1}{3}} + 10 \cdot 10) + 900 (\frac{x^{\frac{1}{3}} + 10}{3}) - 1000$

Simplify and combine like terms.

Simplify each term.

Multiply $x^{\frac{1}{3}}$ by $x^{\frac{1}{3}}$ by adding the exponents.

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

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 30 x^{\frac{2}{3}} + 300 x^{\frac{1}{3}} + 1000 - 30 (x^{\frac{1}{3} + \frac{1}{3}} + x^{\frac{1}{3}} \cdot 10 + 10 x^{\frac{1}{3}} + 10 \cdot 10) + 900 (\frac{x^{\frac{1}{3}} + 10}{3}) - 1000$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 30 x^{\frac{2}{3}} + 300 x^{\frac{1}{3}} + 1000 - 30 (x^{\frac{1 + 1}{3}} + x^{\frac{1}{3}} \cdot 10 + 10 x^{\frac{1}{3}} + 10 \cdot 10) + 900 (\frac{x^{\frac{1}{3}} + 10}{3}) - 1000$

Add $1$ and $1$ .

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 30 x^{\frac{2}{3}} + 300 x^{\frac{1}{3}} + 1000 - 30 (x^{\frac{2}{3}} + x^{\frac{1}{3}} \cdot 10 + 10 x^{\frac{1}{3}} + 10 \cdot 10) + 900 (\frac{x^{\frac{1}{3}} + 10}{3}) - 1000$

Move $10$ to the left of $x^{\frac{1}{3}}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 30 x^{\frac{2}{3}} + 300 x^{\frac{1}{3}} + 1000 - 30 (x^{\frac{2}{3}} + 10 \cdot x^{\frac{1}{3}} + 10 x^{\frac{1}{3}} + 10 \cdot 10) + 900 (\frac{x^{\frac{1}{3}} + 10}{3}) - 1000$

Multiply $10$ by $10$ .

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 30 x^{\frac{2}{3}} + 300 x^{\frac{1}{3}} + 1000 - 30 (x^{\frac{2}{3}} + 10 x^{\frac{1}{3}} + 10 x^{\frac{1}{3}} + 100) + 900 (\frac{x^{\frac{1}{3}} + 10}{3}) - 1000$

Add $10 x^{\frac{1}{3}}$ and $10 x^{\frac{1}{3}}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 30 x^{\frac{2}{3}} + 300 x^{\frac{1}{3}} + 1000 - 30 (x^{\frac{2}{3}} + 20 x^{\frac{1}{3}} + 100) + 900 (\frac{x^{\frac{1}{3}} + 10}{3}) - 1000$

Apply the distributive property.

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 30 x^{\frac{2}{3}} + 300 x^{\frac{1}{3}} + 1000 - 30 x^{\frac{2}{3}} - 30 (20 x^{\frac{1}{3}}) - 30 \cdot 100 + 900 (\frac{x^{\frac{1}{3}} + 10}{3}) - 1000$

Simplify.

Multiply $20$ by $- 30$ .

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 30 x^{\frac{2}{3}} + 300 x^{\frac{1}{3}} + 1000 - 30 x^{\frac{2}{3}} - 600 x^{\frac{1}{3}} - 30 \cdot 100 + 900 (\frac{x^{\frac{1}{3}} + 10}{3}) - 1000$

Multiply $- 30$ by $100$ .

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 30 x^{\frac{2}{3}} + 300 x^{\frac{1}{3}} + 1000 - 30 x^{\frac{2}{3}} - 600 x^{\frac{1}{3}} - 3000 + 900 (\frac{x^{\frac{1}{3}} + 10}{3}) - 1000$

Cancel the common factor of $3$ .

Factor $3$ out of $900$ .

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 30 x^{\frac{2}{3}} + 300 x^{\frac{1}{3}} + 1000 - 30 x^{\frac{2}{3}} - 600 x^{\frac{1}{3}} - 3000 + 3 (300) (\frac{x^{\frac{1}{3}} + 10}{3}) - 1000$

Cancel the common factor.

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 30 x^{\frac{2}{3}} + 300 x^{\frac{1}{3}} + 1000 - 30 x^{\frac{2}{3}} - 600 x^{\frac{1}{3}} - 3000 + 3 \cdot (300 (\frac{x^{\frac{1}{3}} + 10}{3})) - 1000$

Rewrite the expression.

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 30 x^{\frac{2}{3}} + 300 x^{\frac{1}{3}} + 1000 - 30 x^{\frac{2}{3}} - 600 x^{\frac{1}{3}} - 3000 + 300 (x^{\frac{1}{3}} + 10) - 1000$

Apply the distributive property.

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 30 x^{\frac{2}{3}} + 300 x^{\frac{1}{3}} + 1000 - 30 x^{\frac{2}{3}} - 600 x^{\frac{1}{3}} - 3000 + 300 x^{\frac{1}{3}} + 300 \cdot 10 - 1000$

Multiply $300$ by $10$ .

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 30 x^{\frac{2}{3}} + 300 x^{\frac{1}{3}} + 1000 - 30 x^{\frac{2}{3}} - 600 x^{\frac{1}{3}} - 3000 + 300 x^{\frac{1}{3}} + 3000 - 1000$

Simplify by adding terms.

Combine the opposite terms in $x + 30 x^{\frac{2}{3}} + 300 x^{\frac{1}{3}} + 1000 - 30 x^{\frac{2}{3}} - 600 x^{\frac{1}{3}} - 3000 + 300 x^{\frac{1}{3}} + 3000 - 1000$ .

Subtract $30 x^{\frac{2}{3}}$ from $30 x^{\frac{2}{3}}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 300 x^{\frac{1}{3}} + 1000 + 0 - 600 x^{\frac{1}{3}} - 3000 + 300 x^{\frac{1}{3}} + 3000 - 1000$

Add $x + 300 x^{\frac{1}{3}} + 1000$ and $0$ .

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 300 x^{\frac{1}{3}} + 1000 - 600 x^{\frac{1}{3}} - 3000 + 300 x^{\frac{1}{3}} + 3000 - 1000$

Add $- 3000$ and $3000$ .

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 300 x^{\frac{1}{3}} + 1000 - 600 x^{\frac{1}{3}} + 300 x^{\frac{1}{3}} + 0 - 1000$

Add $x + 300 x^{\frac{1}{3}} + 1000 - 600 x^{\frac{1}{3}} + 300 x^{\frac{1}{3}}$ and $0$ .

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 300 x^{\frac{1}{3}} + 1000 - 600 x^{\frac{1}{3}} + 300 x^{\frac{1}{3}} - 1000$

Subtract $1000$ from $1000$ .

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 300 x^{\frac{1}{3}} - 600 x^{\frac{1}{3}} + 300 x^{\frac{1}{3}} + 0$

Add $x + 300 x^{\frac{1}{3}} - 600 x^{\frac{1}{3}} + 300 x^{\frac{1}{3}}$ and $0$ .

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 300 x^{\frac{1}{3}} - 600 x^{\frac{1}{3}} + 300 x^{\frac{1}{3}}$

Subtract $600 x^{\frac{1}{3}}$ from $300 x^{\frac{1}{3}}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x - 300 x^{\frac{1}{3}} + 300 x^{\frac{1}{3}}$

Combine the opposite terms in $x - 300 x^{\frac{1}{3}} + 300 x^{\frac{1}{3}}$ .

Add $- 300 x^{\frac{1}{3}}$ and $300 x^{\frac{1}{3}}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (\frac{x^{\frac{1}{3}} + 10}{3}) = x$

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

Set up the composite result function.

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

Evaluate $f (27 x^{3} - 270 x^{2} + 900 x - 1000)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (27 x^{3} - 270 x^{2} + 900 x - 1000) = \frac{{(27 x^{3} - 270 x^{2} + 900 x - 1000)}^{\frac{1}{3}} + 10}{3}$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 27 x^{3} - 270 x^{2} + 900 x - 1000$ is the inverse of $f (x) = \frac{x^{\frac{1}{3}} + 10}{3}$ .

$f^{- 1} (x) = 27 x^{3} - 270 x^{2} + 900 x - 1000$

Find the Inverse f(x)=(x^(1/3)+10)/3

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