Find the Inverse y=x^3

$y = x^{3}$

Interchange the variables.

$x = y^{3}$

Solve for $y$ .

Rewrite the equation as $y^{3} = x$ .

$y^{3} = x$

Take the cube root of both sides of the equation to eliminate the exponent on the left side.

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

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

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

Verify if $f^{- 1} (x) = \sqrt[3]{x}$ is the inverse of $f (x) = x^{3}$ .

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

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

Remove parentheses.

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

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

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

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

Set up the composite result function.

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

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

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

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

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

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

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

$f (\sqrt[3]{x}) = x^{\frac{1}{3} \cdot 3}$

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

$f (\sqrt[3]{x}) = x^{\frac{3}{3}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$f (\sqrt[3]{x}) = x^{\frac{3}{3}}$

Rewrite the expression.

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

Simplify.

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

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

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