Precalculus Examples

$f (x) = 4 \sqrt[3]{x}$ ; find $f^{- 1} (x)$

Step 1

Write $f (x) = 4 \sqrt[3]{x}$ as an equation.

$y = 4 \sqrt[3]{x}$

Step 2

Interchange the variables.

$x = 4 \sqrt[3]{y}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $4 \sqrt[3]{y} = x$ .

$4 \sqrt[3]{y} = x$

Step 3.2

Divide each term in $4 \sqrt[3]{y} = x$ by $4$ and simplify.

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Step 3.2.1

Divide each term in $4 \sqrt[3]{y} = x$ by $4$ .

$\frac{4 \sqrt[3]{y}}{4} = \frac{x}{4}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $4$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{4 \sqrt[3]{y}}{4} = \frac{x}{4}$

Step 3.2.2.1.2

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

$\sqrt[3]{y} = \frac{x}{4}$

Step 3.3

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

${\sqrt[3]{y}}^{3} = {(\frac{x}{4})}^{3}$

Step 3.4

Simplify each side of the equation.

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Step 3.4.1

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

${(y^{\frac{1}{3}})}^{3} = {(\frac{x}{4})}^{3}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

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

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Step 3.4.2.1.1

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

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Step 3.4.2.1.1.1

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

$y^{\frac{1}{3} \cdot 3} = {(\frac{x}{4})}^{3}$

Step 3.4.2.1.1.2

Cancel the common factor of $3$ .

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Step 3.4.2.1.1.2.1

Cancel the common factor.

$y^{\frac{1}{3} \cdot 3} = {(\frac{x}{4})}^{3}$

Step 3.4.2.1.1.2.2

Rewrite the expression.

$y^{1} = {(\frac{x}{4})}^{3}$

Step 3.4.2.1.2

Simplify.

$y = {(\frac{x}{4})}^{3}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Simplify ${(\frac{x}{4})}^{3}$ .

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Step 3.4.3.1.1

Apply the product rule to $\frac{x}{4}$ .

$y = \frac{x^{3}}{4^{3}}$

Step 3.4.3.1.2

Raise $4$ to the power of $3$ .

$y = \frac{x^{3}}{64}$

Step 4

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

$f^{- 1} (x) = \frac{x^{3}}{64}$

Step 5

Verify if $f^{- 1} (x) = \frac{x^{3}}{64}$ is the inverse of $f (x) = 4 \sqrt[3]{x}$ .

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Step 5.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $f^{- 1} (4 \sqrt[3]{x})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (4 \sqrt[3]{x}) = \frac{{(4 \sqrt[3]{x})}^{3}}{64}$

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

Apply the product rule to $4 \sqrt[3]{x}$ .

$f^{- 1} (4 \sqrt[3]{x}) = \frac{4^{3} {\sqrt[3]{x}}^{3}}{64}$

Step 5.2.3.2

Raise $4$ to the power of $3$ .

$f^{- 1} (4 \sqrt[3]{x}) = \frac{64 {\sqrt[3]{x}}^{3}}{64}$

Step 5.2.3.3

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

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Step 5.2.3.3.1

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

$f^{- 1} (4 \sqrt[3]{x}) = \frac{64 {(x^{\frac{1}{3}})}^{3}}{64}$

Step 5.2.3.3.2

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

$f^{- 1} (4 \sqrt[3]{x}) = \frac{64 x^{\frac{1}{3} \cdot 3}}{64}$

Step 5.2.3.3.3

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

$f^{- 1} (4 \sqrt[3]{x}) = \frac{64 x^{\frac{3}{3}}}{64}$

Step 5.2.3.3.4

Cancel the common factor of $3$ .

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Step 5.2.3.3.4.1

Cancel the common factor.

$f^{- 1} (4 \sqrt[3]{x}) = \frac{64 x^{\frac{3}{3}}}{64}$

Step 5.2.3.3.4.2

Rewrite the expression.

$f^{- 1} (4 \sqrt[3]{x}) = \frac{64 x}{64}$

Step 5.2.3.3.5

Simplify.

$f^{- 1} (4 \sqrt[3]{x}) = \frac{64 x}{64}$

Step 5.2.4

Cancel the common factor of $64$ .

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Step 5.2.4.1

Cancel the common factor.

$f^{- 1} (4 \sqrt[3]{x}) = \frac{64 x}{64}$

Step 5.2.4.2

Divide $x$ by $1$ .

$f^{- 1} (4 \sqrt[3]{x}) = x$

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $f (\frac{x^{3}}{64})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x^{3}}{64}) = 4 \sqrt[3]{\frac{x^{3}}{64}}$

Step 5.3.3

Remove parentheses.

$f (\frac{x^{3}}{64}) = 4 \sqrt[3]{\frac{x^{3}}{64}}$

Step 5.3.4

Rewrite $64$ as $4^{3}$ .

$f (\frac{x^{3}}{64}) = 4 \sqrt[3]{\frac{x^{3}}{4^{3}}}$

Step 5.3.5

Rewrite $\frac{x^{3}}{4^{3}}$ as ${(\frac{x}{4})}^{3}$ .

$f (\frac{x^{3}}{64}) = 4 \sqrt[3]{{(\frac{x}{4})}^{3}}$

Step 5.3.6

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

$f (\frac{x^{3}}{64}) = 4 (\frac{x}{4})$

Step 5.3.7

Cancel the common factor of $4$ .

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Step 5.3.7.1

Cancel the common factor.

$f (\frac{x^{3}}{64}) = 4 (\frac{x}{4})$

Step 5.3.7.2

Rewrite the expression.

$f (\frac{x^{3}}{64}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x^{3}}{64}$ is the inverse of $f (x) = 4 \sqrt[3]{x}$ .

$f^{- 1} (x) = \frac{x^{3}}{64}$