Algebra Examples

$y = \frac{1}{4} x^{3} - 2$

Step 1

Interchange the variables.

$x = \frac{1}{4} y^{3} - 2$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{1}{4} y^{3} - 2 = x$ .

$\frac{1}{4} y^{3} - 2 = x$

Step 2.2

Combine $\frac{1}{4}$ and $y^{3}$ .

$\frac{y^{3}}{4} - 2 = x$

Step 2.3

Add $2$ to both sides of the equation.

$\frac{y^{3}}{4} = x + 2$

Step 2.4

Multiply both sides of the equation by $4$ .

$4 \frac{y^{3}}{4} = 4 (x + 2)$

Step 2.5

Simplify both sides of the equation.

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Step 2.5.1

Simplify the left side.

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Step 2.5.1.1

Cancel the common factor of $4$ .

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Step 2.5.1.1.1

Cancel the common factor.

$4 \frac{y^{3}}{4} = 4 (x + 2)$

Step 2.5.1.1.2

Rewrite the expression.

$y^{3} = 4 (x + 2)$

Step 2.5.2

Simplify the right side.

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Step 2.5.2.1

Simplify $4 (x + 2)$ .

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Step 2.5.2.1.1

Apply the distributive property.

$y^{3} = 4 x + 4 \cdot 2$

Step 2.5.2.1.2

Multiply $4$ by $2$ .

$y^{3} = 4 x + 8$

Step 2.6

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

$y = \sqrt[3]{4 x + 8}$

Step 2.7

Factor $4$ out of $4 x + 8$ .

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Step 2.7.1

Factor $4$ out of $4 x$ .

$y = \sqrt[3]{4 (x) + 8}$

Step 2.7.2

Factor $4$ out of $8$ .

$y = \sqrt[3]{4 x + 4 \cdot 2}$

Step 2.7.3

Factor $4$ out of $4 x + 4 \cdot 2$ .

$y = \sqrt[3]{4 (x + 2)}$

Step 3

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

$f^{- 1} (x) = \sqrt[3]{4 (x + 2)}$

Step 4

Verify if $f^{- 1} (x) = \sqrt[3]{4 (x + 2)}$ is the inverse of $f (x) = \frac{1}{4} x^{3} - 2$ .

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Step 4.1

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

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

Evaluate $f^{- 1} (\frac{1}{4} x^{3} - 2)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{1}{4} x^{3} - 2) = \sqrt[3]{4 ((\frac{1}{4} x^{3} - 2) + 2)}$

Step 4.2.3

Combine $\frac{1}{4}$ and $x^{3}$ .

$f^{- 1} (\frac{1}{4} x^{3} - 2) = \sqrt[3]{4 (\frac{x^{3}}{4} - 2 + 2)}$

Step 4.2.4

Simplify by adding numbers.

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Step 4.2.4.1

Add $- 2$ and $2$ .

$f^{- 1} (\frac{1}{4} x^{3} - 2) = \sqrt[3]{4 (\frac{x^{3}}{4} + 0)}$

Step 4.2.4.2

Add $\frac{x^{3}}{4}$ and $0$ .

$f^{- 1} (\frac{1}{4} x^{3} - 2) = \sqrt[3]{4 (\frac{x^{3}}{4})}$

Step 4.2.5

Combine $4$ and $\frac{x^{3}}{4}$ .

$f^{- 1} (\frac{1}{4} x^{3} - 2) = \sqrt[3]{\frac{4 x^{3}}{4}}$

Step 4.2.6

Reduce the expression by cancelling the common factors.

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Step 4.2.6.1

Reduce the expression $\frac{4 x^{3}}{4}$ by cancelling the common factors.

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Step 4.2.6.1.1

Cancel the common factor.

$f^{- 1} (\frac{1}{4} x^{3} - 2) = \sqrt[3]{\frac{4 x^{3}}{4}}$

Step 4.2.6.1.2

Rewrite the expression.

$f^{- 1} (\frac{1}{4} x^{3} - 2) = \sqrt[3]{\frac{x^{3}}{1}}$

Step 4.2.6.2

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

$f^{- 1} (\frac{1}{4} x^{3} - 2) = \sqrt[3]{x^{3}}$

Step 4.2.7

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

$f^{- 1} (\frac{1}{4} x^{3} - 2) = x$

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

Evaluate $f (\sqrt[3]{4 (x + 2)})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\sqrt[3]{4 (x + 2)}) = \frac{1}{4} \cdot {(\sqrt[3]{4 (x + 2)})}^{3} - 2$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Rewrite ${\sqrt[3]{4 (x + 2)}}^{3}$ as $4 (x + 2)$ .

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Step 4.3.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4 (x + 2)}$ as ${(4 (x + 2))}^{\frac{1}{3}}$ .

$f (\sqrt[3]{4 (x + 2)}) = \frac{1}{4} \cdot {({(4 (x + 2))}^{\frac{1}{3}})}^{3} - 2$

Step 4.3.3.1.2

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

$f (\sqrt[3]{4 (x + 2)}) = \frac{1}{4} \cdot {(4 (x + 2))}^{\frac{1}{3} \cdot 3} - 2$

Step 4.3.3.1.3

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

$f (\sqrt[3]{4 (x + 2)}) = \frac{1}{4} \cdot {(4 (x + 2))}^{\frac{3}{3}} - 2$

Step 4.3.3.1.4

Cancel the common factor of $3$ .

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Step 4.3.3.1.4.1

Cancel the common factor.

$f (\sqrt[3]{4 (x + 2)}) = \frac{1}{4} \cdot {(4 (x + 2))}^{\frac{3}{3}} - 2$

Step 4.3.3.1.4.2

Rewrite the expression.

$f (\sqrt[3]{4 (x + 2)}) = \frac{1}{4} \cdot (4 (x + 2)) - 2$

Step 4.3.3.1.5

Simplify.

$f (\sqrt[3]{4 (x + 2)}) = \frac{1}{4} \cdot (4 (x + 2)) - 2$

Step 4.3.3.2

Cancel the common factor of $4$ .

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Step 4.3.3.2.1

Cancel the common factor.

$f (\sqrt[3]{4 (x + 2)}) = \frac{1}{4} \cdot (4 (x + 2)) - 2$

Step 4.3.3.2.2

Rewrite the expression.

$f (\sqrt[3]{4 (x + 2)}) = x + 2 - 2$

Step 4.3.4

Combine the opposite terms in $x + 2 - 2$ .

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Step 4.3.4.1

Subtract $2$ from $2$ .

$f (\sqrt[3]{4 (x + 2)}) = x + 0$

Step 4.3.4.2

Add $x$ and $0$ .

$f (\sqrt[3]{4 (x + 2)}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \sqrt[3]{4 (x + 2)}$ is the inverse of $f (x) = \frac{1}{4} x^{3} - 2$ .

$f^{- 1} (x) = \sqrt[3]{4 (x + 2)}$