Precalculus Examples

$g (x) = x^{3}$

Step 1

Write $g (x) = x^{3}$ as an equation.

$y = x^{3}$

Step 2

Interchange the variables.

$x = y^{3}$

Step 3

Solve for $y$ .

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Step 3.1

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

$y^{3} = x$

Step 3.2

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

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

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Remove parentheses.

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

Step 5.2.4

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

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

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

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

Step 5.3.3

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

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Step 5.3.3.1

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

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

Step 5.3.3.2

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

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

Step 5.3.3.3

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

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

Step 5.3.3.4

Cancel the common factor of $3$ .

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Step 5.3.3.4.1

Cancel the common factor.

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

Step 5.3.3.4.2

Rewrite the expression.

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

Step 5.3.3.5

Simplify.

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

Step 5.4

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

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