Algebra Examples

$\frac{1}{8} x^{3}$

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{1}{8} y^{3} = x$ .

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

Step 2.2

Multiply both sides of the equation by $8$ .

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

Step 2.3

Simplify the left side.

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Step 2.3.1

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

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Step 2.3.1.1

Combine $\frac{1}{8}$ and $y^{3}$ .

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

Step 2.3.1.2

Cancel the common factor of $8$ .

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Step 2.3.1.2.1

Cancel the common factor.

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

Step 2.3.1.2.2

Rewrite the expression.

$y^{3} = 8 x$

Step 2.4

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

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

Step 2.5

Simplify $\sqrt[3]{8 x}$ .

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Step 2.5.1

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

$y = \sqrt[3]{2^{3} x}$

Step 2.5.2

Pull terms out from under the radical.

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

Step 3

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

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

Step 4

Verify if $f^{- 1} (x) = 2 \sqrt[3]{x}$ is the inverse of $f (x) = \frac{1}{8} x^{3}$ .

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

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

Step 4.2.3

Remove parentheses.

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

Step 4.2.4

Combine $\frac{1}{8}$ and $x^{3}$ .

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

Step 4.2.5

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

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

Step 4.2.6

Rewrite $\frac{x^{3}}{2^{3}}$ as ${(\frac{x}{2})}^{3}$ .

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

Step 4.2.7

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

$f^{- 1} (\frac{1}{8} x^{3}) = 2 (\frac{x}{2})$

Step 4.2.8

Cancel the common factor of $2$ .

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Step 4.2.8.1

Cancel the common factor.

$f^{- 1} (\frac{1}{8} x^{3}) = 2 (\frac{x}{2})$

Step 4.2.8.2

Rewrite the expression.

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

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

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

Step 4.3.3

Simplify the expression.

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Step 4.3.3.1

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

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

Step 4.3.3.2

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

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

Step 4.3.4

Cancel the common factor of $8$ .

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Step 4.3.4.1

Factor $8$ out of $8 {\sqrt[3]{x}}^{3}$ .

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

Step 4.3.4.2

Cancel the common factor.

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

Step 4.3.4.3

Rewrite the expression.

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

Step 4.3.5

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

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Step 4.3.5.1

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

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

Step 4.3.5.2

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

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

Step 4.3.5.3

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

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

Step 4.3.5.4

Cancel the common factor of $3$ .

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Step 4.3.5.4.1

Cancel the common factor.

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

Step 4.3.5.4.2

Rewrite the expression.

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

Step 4.3.5.5

Simplify.

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

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 2 \sqrt[3]{x}$ is the inverse of $f (x) = \frac{1}{8} x^{3}$ .

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