Find the Inverse f(x) = cube root of 6x

$f (x) = \sqrt[3]{6 x}$

Step 1

Write $f (x) = \sqrt[3]{6 x}$ as an equation.

$y = \sqrt[3]{6 x}$

Step 2

Interchange the variables.

$x = \sqrt[3]{6 y}$

Step 3

Solve for $y$ .

Rewrite the equation as $\sqrt[3]{6 y} = x$ .

$\sqrt[3]{6 y} = x$

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

${\sqrt[3]{6 y}}^{3} = x^{3}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{6 y}$ as ${(6 y)}^{\frac{1}{3}}$ .

${({(6 y)}^{\frac{1}{3}})}^{3} = x^{3}$

Simplify the left side.

Simplify ${({(6 y)}^{\frac{1}{3}})}^{3}$ .

Multiply the exponents in ${({(6 y)}^{\frac{1}{3}})}^{3}$ .

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

${(6 y)}^{\frac{1}{3} \cdot 3} = x^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(6 y)}^{\frac{1}{3} \cdot 3} = x^{3}$

Rewrite the expression.

${(6 y)}^{1} = x^{3}$

Simplify.

$6 y = x^{3}$

Divide each term in $6 y = x^{3}$ by $6$ and simplify.

Divide each term in $6 y = x^{3}$ by $6$ .

$\frac{6 y}{6} = \frac{x^{3}}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 y}{6} = \frac{x^{3}}{6}$

Divide $y$ by $1$ .

$y = \frac{x^{3}}{6}$

Step 4

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

$f^{- 1} (x) = \frac{x^{3}}{6}$

Step 5

Verify if $f^{- 1} (x) = \frac{x^{3}}{6}$ is the inverse of $f (x) = \sqrt[3]{6 x}$ .

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 $f^{- 1} (\sqrt[3]{6 x})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\sqrt[3]{6 x}) = \frac{{(\sqrt[3]{6 x})}^{3}}{6}$

Rewrite ${\sqrt[3]{6 x}}^{3}$ as $6 x$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{6 x}$ as ${(6 x)}^{\frac{1}{3}}$ .

$f^{- 1} (\sqrt[3]{6 x}) = \frac{{({(6 x)}^{\frac{1}{3}})}^{3}}{6}$

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

$f^{- 1} (\sqrt[3]{6 x}) = \frac{{(6 x)}^{\frac{1}{3} \cdot 3}}{6}$

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

$f^{- 1} (\sqrt[3]{6 x}) = \frac{{(6 x)}^{\frac{3}{3}}}{6}$

Cancel the common factor of $3$ .

Cancel the common factor.

$f^{- 1} (\sqrt[3]{6 x}) = \frac{{(6 x)}^{\frac{3}{3}}}{6}$

Rewrite the expression.

$f^{- 1} (\sqrt[3]{6 x}) = \frac{6 x}{6}$

Simplify.

$f^{- 1} (\sqrt[3]{6 x}) = \frac{6 x}{6}$

Cancel the common factor of $6$ .

Cancel the common factor.

$f^{- 1} (\sqrt[3]{6 x}) = \frac{6 x}{6}$

Divide $x$ by $1$ .

$f^{- 1} (\sqrt[3]{6 x}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{x^{3}}{6})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x^{3}}{6}) = \sqrt[3]{6 (\frac{x^{3}}{6})}$

Combine $6$ and $\frac{x^{3}}{6}$ .

$f (\frac{x^{3}}{6}) = \sqrt[3]{\frac{6 x^{3}}{6}}$

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{6 x^{3}}{6}$ by cancelling the common factors.

Cancel the common factor.

$f (\frac{x^{3}}{6}) = \sqrt[3]{\frac{6 x^{3}}{6}}$

Rewrite the expression.

$f (\frac{x^{3}}{6}) = \sqrt[3]{\frac{x^{3}}{1}}$

Divide $x^{3}$ by $1$ .

$f (\frac{x^{3}}{6}) = \sqrt[3]{x^{3}}$

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

$f (\frac{x^{3}}{6}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x^{3}}{6}$ is the inverse of $f (x) = \sqrt[3]{6 x}$ .

$f^{- 1} (x) = \frac{x^{3}}{6}$

Find the Inverse f(x) = cube root of 6x

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