Algebra Examples

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

Step 1

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

$y = x^{3}$

Step 2

Interchange the variables.

$x = y^{3}$

Step 3

Solve for $y$ .

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

$y^{3} = x$

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

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

Step 4

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

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

Step 5

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

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

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

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Set up the composite result function.

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

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

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

Remove parentheses.

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

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

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

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

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Set up the composite result function.

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

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

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

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

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

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

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

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

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

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

Cancel the common factor of $3$ .

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Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

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

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