Precalculus Examples

$g (x) = \sqrt[3]{8 x}$

Step 1

Write $g (x) = \sqrt[3]{8 x}$ as an equation.

$y = \sqrt[3]{8 x}$

Step 2

Interchange the variables.

$x = \sqrt[3]{8 y}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\sqrt[3]{8 y} = x$ .

$\sqrt[3]{8 y} = x$

Step 3.2

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

${\sqrt[3]{8 y}}^{3} = x^{3}$

Step 3.3

Simplify each side of the equation.

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Step 3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{8 y}$ as ${(8 y)}^{\frac{1}{3}}$ .

${({(8 y)}^{\frac{1}{3}})}^{3} = x^{3}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Simplify ${({(8 y)}^{\frac{1}{3}})}^{3}$ .

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Step 3.3.2.1.1

Multiply the exponents in ${({(8 y)}^{\frac{1}{3}})}^{3}$ .

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Step 3.3.2.1.1.1

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

${(8 y)}^{\frac{1}{3} \cdot 3} = x^{3}$

Step 3.3.2.1.1.2

Cancel the common factor of $3$ .

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Step 3.3.2.1.1.2.1

Cancel the common factor.

${(8 y)}^{\frac{1}{3} \cdot 3} = x^{3}$

Step 3.3.2.1.1.2.2

Rewrite the expression.

${(8 y)}^{1} = x^{3}$

Step 3.3.2.1.2

Simplify.

$8 y = x^{3}$

Step 3.4

Divide each term in $8 y = x^{3}$ by $8$ and simplify.

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Step 3.4.1

Divide each term in $8 y = x^{3}$ by $8$ .

$\frac{8 y}{8} = \frac{x^{3}}{8}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $8$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{8 y}{8} = \frac{x^{3}}{8}$

Step 3.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{x^{3}}{8}$

Step 4

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

$g^{- 1} (x) = \frac{x^{3}}{8}$

Step 5

Verify if $g^{- 1} (x) = \frac{x^{3}}{8}$ is the inverse of $g (x) = \sqrt[3]{8 x}$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $g^{- 1} (\sqrt[3]{8 x})$ by substituting in the value of $g$ into $g^{- 1}$ .

$g^{- 1} (\sqrt[3]{8 x}) = \frac{{(\sqrt[3]{8 x})}^{3}}{8}$

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

Rewrite $8$ as $2^{3}$ .

$g^{- 1} (\sqrt[3]{8 x}) = \frac{{\sqrt[3]{2^{3} x}}^{3}}{8}$

Step 5.2.3.2

Pull terms out from under the radical.

$g^{- 1} (\sqrt[3]{8 x}) = \frac{{(2 \sqrt[3]{x})}^{3}}{8}$

Step 5.2.3.3

Apply the product rule to $2 \sqrt[3]{x}$ .

$g^{- 1} (\sqrt[3]{8 x}) = \frac{2^{3} {\sqrt[3]{x}}^{3}}{8}$

Step 5.2.3.4

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

$g^{- 1} (\sqrt[3]{8 x}) = \frac{8 {\sqrt[3]{x}}^{3}}{8}$

Step 5.2.3.5

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

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Step 5.2.3.5.1

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

$g^{- 1} (\sqrt[3]{8 x}) = \frac{8 {(x^{\frac{1}{3}})}^{3}}{8}$

Step 5.2.3.5.2

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

$g^{- 1} (\sqrt[3]{8 x}) = \frac{8 x^{\frac{1}{3} \cdot 3}}{8}$

Step 5.2.3.5.3

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

$g^{- 1} (\sqrt[3]{8 x}) = \frac{8 x^{\frac{3}{3}}}{8}$

Step 5.2.3.5.4

Cancel the common factor of $3$ .

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Step 5.2.3.5.4.1

Cancel the common factor.

$g^{- 1} (\sqrt[3]{8 x}) = \frac{8 x^{\frac{3}{3}}}{8}$

Step 5.2.3.5.4.2

Rewrite the expression.

$g^{- 1} (\sqrt[3]{8 x}) = \frac{8 x}{8}$

Step 5.2.3.5.5

Simplify.

$g^{- 1} (\sqrt[3]{8 x}) = \frac{8 x}{8}$

Step 5.2.4

Cancel the common factor of $8$ .

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Step 5.2.4.1

Cancel the common factor.

$g^{- 1} (\sqrt[3]{8 x}) = \frac{8 x}{8}$

Step 5.2.4.2

Divide $x$ by $1$ .

$g^{- 1} (\sqrt[3]{8 x}) = x$

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $g (\frac{x^{3}}{8})$ by substituting in the value of $g^{- 1}$ into $g$ .

$g (\frac{x^{3}}{8}) = \sqrt[3]{8 (\frac{x^{3}}{8})}$

Step 5.3.3

Combine $8$ and $\frac{x^{3}}{8}$ .

$g (\frac{x^{3}}{8}) = \sqrt[3]{\frac{8 x^{3}}{8}}$

Step 5.3.4

Reduce the expression by cancelling the common factors.

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Step 5.3.4.1

Reduce the expression $\frac{8 x^{3}}{8}$ by cancelling the common factors.

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Step 5.3.4.1.1

Cancel the common factor.

$g (\frac{x^{3}}{8}) = \sqrt[3]{\frac{8 x^{3}}{8}}$

Step 5.3.4.1.2

Rewrite the expression.

$g (\frac{x^{3}}{8}) = \sqrt[3]{\frac{x^{3}}{1}}$

Step 5.3.4.2

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

$g (\frac{x^{3}}{8}) = \sqrt[3]{x^{3}}$

Step 5.3.5

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

$g (\frac{x^{3}}{8}) = x$

Step 5.4

Since $g^{- 1} (g (x)) = x$ and $g (g^{- 1} (x)) = x$ , then $g^{- 1} (x) = \frac{x^{3}}{8}$ is the inverse of $g (x) = \sqrt[3]{8 x}$ .

$g^{- 1} (x) = \frac{x^{3}}{8}$