Find the Inverse f(x)=3 cube root of x

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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

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

Divide each term in $3 \sqrt[3]{y} = x$ by $3$ and simplify.

Divide each term in $3 \sqrt[3]{y} = x$ by $3$ .

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $\sqrt[3]{y}$ by $1$ .

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

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

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

Simplify each side of the equation.

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

Simplify ${(\frac{x}{3})}^{3}$ .

Apply the product rule to $\frac{x}{3}$ .

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

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

$y = \frac{x^{3}}{27}$

Step 4

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

$f^{- 1} (x) = \frac{x^{3}}{27}$

Step 5

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

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

$f^{- 1} (3 \sqrt[3]{x}) = \frac{{(3 \sqrt[3]{x})}^{3}}{27}$

Simplify the numerator.

Apply the product rule to $3 \sqrt[3]{x}$ .

$f^{- 1} (3 \sqrt[3]{x}) = \frac{3^{3} {\sqrt[3]{x}}^{3}}{27}$

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

$f^{- 1} (3 \sqrt[3]{x}) = \frac{27 {\sqrt[3]{x}}^{3}}{27}$

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

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

$f^{- 1} (3 \sqrt[3]{x}) = \frac{27 {(x^{\frac{1}{3}})}^{3}}{27}$

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

$f^{- 1} (3 \sqrt[3]{x}) = \frac{27 x^{\frac{1}{3} \cdot 3}}{27}$

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

$f^{- 1} (3 \sqrt[3]{x}) = \frac{27 x^{\frac{3}{3}}}{27}$

Cancel the common factor of $3$ .

Cancel the common factor.

$f^{- 1} (3 \sqrt[3]{x}) = \frac{27 x^{\frac{3}{3}}}{27}$

Rewrite the expression.

$f^{- 1} (3 \sqrt[3]{x}) = \frac{27 x}{27}$

Simplify.

$f^{- 1} (3 \sqrt[3]{x}) = \frac{27 x}{27}$

Cancel the common factor of $27$ .

Cancel the common factor.

$f^{- 1} (3 \sqrt[3]{x}) = \frac{27 x}{27}$

Divide $x$ by $1$ .

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

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

Set up the composite result function.

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

Evaluate $f (\frac{x^{3}}{27})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x^{3}}{27}) = 3 \sqrt[3]{\frac{x^{3}}{27}}$

Remove parentheses.

$f (\frac{x^{3}}{27}) = 3 \sqrt[3]{\frac{x^{3}}{27}}$

Rewrite $27$ as $3^{3}$ .

$f (\frac{x^{3}}{27}) = 3 \sqrt[3]{\frac{x^{3}}{3^{3}}}$

Rewrite $\frac{x^{3}}{3^{3}}$ as ${(\frac{x}{3})}^{3}$ .

$f (\frac{x^{3}}{27}) = 3 \sqrt[3]{{(\frac{x}{3})}^{3}}$

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

$f (\frac{x^{3}}{27}) = 3 (\frac{x}{3})$

Cancel the common factor of $3$ .

Cancel the common factor.

$f (\frac{x^{3}}{27}) = 3 (\frac{x}{3})$

Rewrite the expression.

$f (\frac{x^{3}}{27}) = x$

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

$f^{- 1} (x) = \frac{x^{3}}{27}$

Find the Inverse f(x)=3 cube root of x

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