Finite Math Examples

$s (x) = x (x^{2} - \frac{8}{x})$

Step 1

Write $s (x) = x (x^{2} - \frac{8}{x})$ as an equation.

$y = x (x^{2} - \frac{8}{x})$

Step 2

Interchange the variables.

$x = y (y^{2} - \frac{8}{y})$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $y (y^{2} - \frac{8}{y}) = x$ .

$y (y^{2} - \frac{8}{y}) = x$

Step 3.2

Simplify $y (y^{2} - \frac{8}{y})$ .

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Step 3.2.1

Apply the distributive property.

$y \cdot y^{2} + y (- \frac{8}{y}) = x$

Step 3.2.2

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 3.2.2.1

Multiply $y$ by $y^{2}$ .

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Step 3.2.2.1.1

Raise $y$ to the power of $1$ .

$y^{1} y^{2} + y (- \frac{8}{y}) = x$

Step 3.2.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y^{1 + 2} + y (- \frac{8}{y}) = x$

Step 3.2.2.2

Add $1$ and $2$ .

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

Step 3.2.3

Rewrite using the commutative property of multiplication.

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

Step 3.2.4

Simplify each term.

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Step 3.2.4.1

Cancel the common factor of $y$ .

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Step 3.2.4.1.1

Factor $y$ out of $- y$ .

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

Step 3.2.4.1.2

Cancel the common factor.

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

Step 3.2.4.1.3

Rewrite the expression.

$y^{3} - 1 \cdot 8 = x$

Step 3.2.4.2

Multiply $- 1$ by $8$ .

$y^{3} - 8 = x$

Step 3.3

Add $8$ to both sides of the equation.

$y^{3} = x + 8$

Step 3.4

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

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

Step 4

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

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

Step 5

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

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $s^{- 1} (x (x^{2} - \frac{8}{x}))$ by substituting in the value of $s$ into $s^{- 1}$ .

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

Step 5.2.3

Apply the distributive property.

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

Step 5.2.4

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 5.2.4.1

Multiply $x$ by $x^{2}$ .

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Step 5.2.4.1.1

Raise $x$ to the power of $1$ .

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

Step 5.2.4.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 5.2.4.2

Add $1$ and $2$ .

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

Step 5.2.5

Rewrite using the commutative property of multiplication.

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

Step 5.2.6

Simplify each term.

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Step 5.2.6.1

Cancel the common factor of $x$ .

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Step 5.2.6.1.1

Factor $x$ out of $- x$ .

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

Step 5.2.6.1.2

Cancel the common factor.

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

Step 5.2.6.1.3

Rewrite the expression.

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

Step 5.2.6.2

Multiply $- 1$ by $8$ .

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

Step 5.2.7

Simplify by adding numbers.

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Step 5.2.7.1

Add $- 8$ and $8$ .

$s^{- 1} (x (x^{2} - \frac{8}{x})) = \sqrt[3]{x^{3} + 0}$

Step 5.2.7.2

Add $x^{3}$ and $0$ .

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

Step 5.2.8

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

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

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

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

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

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Rewrite ${\sqrt[3]{x + 8}}^{2}$ as $\sqrt[3]{{(x + 8)}^{2}}$ .

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

Step 5.3.3.2

Multiply $\frac{8}{\sqrt[3]{x + 8}}$ by $\frac{{\sqrt[3]{x + 8}}^{2}}{{\sqrt[3]{x + 8}}^{2}}$ .

$s (\sqrt[3]{x + 8}) = \sqrt[3]{x + 8} (\sqrt[3]{{(x + 8)}^{2}} - (\frac{8}{\sqrt[3]{x + 8}} \cdot \frac{{\sqrt[3]{x + 8}}^{2}}{{\sqrt[3]{x + 8}}^{2}}))$

Step 5.3.3.3

Combine and simplify the denominator.

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Step 5.3.3.3.1

Multiply $\frac{8}{\sqrt[3]{x + 8}}$ by $\frac{{\sqrt[3]{x + 8}}^{2}}{{\sqrt[3]{x + 8}}^{2}}$ .

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

Step 5.3.3.3.2

Raise $\sqrt[3]{x + 8}$ to the power of $1$ .

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

Step 5.3.3.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 5.3.3.3.4

Add $1$ and $2$ .

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

Step 5.3.3.3.5

Rewrite ${\sqrt[3]{x + 8}}^{3}$ as $x + 8$ .

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Step 5.3.3.3.5.1

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

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

Step 5.3.3.3.5.2

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

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

Step 5.3.3.3.5.3

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

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

Step 5.3.3.3.5.4

Cancel the common factor of $3$ .

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Step 5.3.3.3.5.4.1

Cancel the common factor.

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

Step 5.3.3.3.5.4.2

Rewrite the expression.

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

Step 5.3.3.3.5.5

Simplify.

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

Step 5.3.3.4

Rewrite ${\sqrt[3]{x + 8}}^{2}$ as $\sqrt[3]{{(x + 8)}^{2}}$ .

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

Step 5.3.4

To write $\sqrt[3]{{(x + 8)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{x + 8}{x + 8}$ .

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

Step 5.3.5

Combine the numerators over the common denominator.

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

Step 5.3.6

Simplify the numerator.

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Step 5.3.6.1

Factor $\sqrt[3]{{(x + 8)}^{2}}$ out of $\sqrt[3]{{(x + 8)}^{2}} (x + 8) - 8 \sqrt[3]{{(x + 8)}^{2}}$ .

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Step 5.3.6.1.1

Factor $\sqrt[3]{{(x + 8)}^{2}}$ out of $- 8 \sqrt[3]{{(x + 8)}^{2}}$ .

$s (\sqrt[3]{x + 8}) = \sqrt[3]{x + 8} (\frac{\sqrt[3]{{(x + 8)}^{2}} (x + 8) + \sqrt[3]{{(x + 8)}^{2}} \cdot - 8}{x + 8})$

Step 5.3.6.1.2

Factor $\sqrt[3]{{(x + 8)}^{2}}$ out of $\sqrt[3]{{(x + 8)}^{2}} (x + 8) + \sqrt[3]{{(x + 8)}^{2}} \cdot - 8$ .

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

Step 5.3.6.2

Subtract $8$ from $8$ .

$s (\sqrt[3]{x + 8}) = \sqrt[3]{x + 8} (\frac{\sqrt[3]{{(x + 8)}^{2}} (x + 0)}{x + 8})$

Step 5.3.6.3

Add $x$ and $0$ .

$s (\sqrt[3]{x + 8}) = \sqrt[3]{x + 8} (\frac{\sqrt[3]{{(x + 8)}^{2}} x}{x + 8})$

Step 5.3.7

Multiply $\sqrt[3]{x + 8} \frac{\sqrt[3]{{(x + 8)}^{2}} x}{x + 8}$ .

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Step 5.3.7.1

Combine $\sqrt[3]{x + 8}$ and $\frac{\sqrt[3]{{(x + 8)}^{2}} x}{x + 8}$ .

$s (\sqrt[3]{x + 8}) = \frac{\sqrt[3]{x + 8} (\sqrt[3]{{(x + 8)}^{2}} x)}{x + 8}$

Step 5.3.7.2

Combine using the product rule for radicals.

$s (\sqrt[3]{x + 8}) = \frac{\sqrt[3]{{(x + 8)}^{2} (x + 8)} x}{x + 8}$

Step 5.3.7.3

Multiply ${(x + 8)}^{2}$ by $x + 8$ by adding the exponents.

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Step 5.3.7.3.1

Multiply ${(x + 8)}^{2}$ by $x + 8$ .

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Step 5.3.7.3.1.1

Raise $x + 8$ to the power of $1$ .

$s (\sqrt[3]{x + 8}) = \frac{\sqrt[3]{{(x + 8)}^{2} (x + 8)} x}{x + 8}$

Step 5.3.7.3.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 5.3.7.3.2

Add $2$ and $1$ .

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

Step 5.3.8

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

$s (\sqrt[3]{x + 8}) = \frac{(x + 8) x}{x + 8}$

Step 5.3.9

Cancel the common factor of $x + 8$ .

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Step 5.3.9.1

Cancel the common factor.

$s (\sqrt[3]{x + 8}) = \frac{(x + 8) x}{x + 8}$

Step 5.3.9.2

Divide $x$ by $1$ .

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

Step 5.4

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

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