Finite Math Examples

$f (x) = 10 x^{3}$

Step 1

Write $f (x) = 10 x^{3}$ as an equation.

$y = 10 x^{3}$

Step 2

Interchange the variables.

$x = 10 y^{3}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $10 y^{3} = x$ .

$10 y^{3} = x$

Step 3.2

Divide each term in $10 y^{3} = x$ by $10$ and simplify.

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Step 3.2.1

Divide each term in $10 y^{3} = x$ by $10$ .

$\frac{10 y^{3}}{10} = \frac{x}{10}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $10$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{10 y^{3}}{10} = \frac{x}{10}$

Step 3.2.2.1.2

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

$y^{3} = \frac{x}{10}$

Step 3.3

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

$y = \sqrt[3]{\frac{x}{10}}$

Step 3.4

Simplify $\sqrt[3]{\frac{x}{10}}$ .

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Step 3.4.1

Rewrite $\sqrt[3]{\frac{x}{10}}$ as $\frac{\sqrt[3]{x}}{\sqrt[3]{10}}$ .

$y = \frac{\sqrt[3]{x}}{\sqrt[3]{10}}$

Step 3.4.2

Multiply $\frac{\sqrt[3]{x}}{\sqrt[3]{10}}$ by $\frac{{\sqrt[3]{10}}^{2}}{{\sqrt[3]{10}}^{2}}$ .

$y = \frac{\sqrt[3]{x}}{\sqrt[3]{10}} \cdot \frac{{\sqrt[3]{10}}^{2}}{{\sqrt[3]{10}}^{2}}$

Step 3.4.3

Combine and simplify the denominator.

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Step 3.4.3.1

Multiply $\frac{\sqrt[3]{x}}{\sqrt[3]{10}}$ by $\frac{{\sqrt[3]{10}}^{2}}{{\sqrt[3]{10}}^{2}}$ .

$y = \frac{\sqrt[3]{x} {\sqrt[3]{10}}^{2}}{\sqrt[3]{10} {\sqrt[3]{10}}^{2}}$

Step 3.4.3.2

Raise $\sqrt[3]{10}$ to the power of $1$ .

$y = \frac{\sqrt[3]{x} {\sqrt[3]{10}}^{2}}{{\sqrt[3]{10}}^{1} {\sqrt[3]{10}}^{2}}$

Step 3.4.3.3

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

$y = \frac{\sqrt[3]{x} {\sqrt[3]{10}}^{2}}{{\sqrt[3]{10}}^{1 + 2}}$

Step 3.4.3.4

Add $1$ and $2$ .

$y = \frac{\sqrt[3]{x} {\sqrt[3]{10}}^{2}}{{\sqrt[3]{10}}^{3}}$

Step 3.4.3.5

Rewrite ${\sqrt[3]{10}}^{3}$ as $10$ .

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Step 3.4.3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{10}$ as $10^{\frac{1}{3}}$ .

$y = \frac{\sqrt[3]{x} {\sqrt[3]{10}}^{2}}{{(10^{\frac{1}{3}})}^{3}}$

Step 3.4.3.5.2

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

$y = \frac{\sqrt[3]{x} {\sqrt[3]{10}}^{2}}{10^{\frac{1}{3} \cdot 3}}$

Step 3.4.3.5.3

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

$y = \frac{\sqrt[3]{x} {\sqrt[3]{10}}^{2}}{10^{\frac{3}{3}}}$

Step 3.4.3.5.4

Cancel the common factor of $3$ .

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Step 3.4.3.5.4.1

Cancel the common factor.

$y = \frac{\sqrt[3]{x} {\sqrt[3]{10}}^{2}}{10^{\frac{3}{3}}}$

Step 3.4.3.5.4.2

Rewrite the expression.

$y = \frac{\sqrt[3]{x} {\sqrt[3]{10}}^{2}}{10^{1}}$

Step 3.4.3.5.5

Evaluate the exponent.

$y = \frac{\sqrt[3]{x} {\sqrt[3]{10}}^{2}}{10}$

Step 3.4.4

Simplify the numerator.

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Step 3.4.4.1

Rewrite ${\sqrt[3]{10}}^{2}$ as $\sqrt[3]{10^{2}}$ .

$y = \frac{\sqrt[3]{x} \sqrt[3]{10^{2}}}{10}$

Step 3.4.4.2

Raise $10$ to the power of $2$ .

$y = \frac{\sqrt[3]{x} \sqrt[3]{100}}{10}$

Step 3.4.5

Simplify with factoring out.

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Step 3.4.5.1

Combine using the product rule for radicals.

$y = \frac{\sqrt[3]{x \cdot 100}}{10}$

Step 3.4.5.2

Reorder factors in $\frac{\sqrt[3]{x \cdot 100}}{10}$ .

$y = \frac{\sqrt[3]{100 x}}{10}$

Step 4

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

$f^{- 1} (x) = \frac{\sqrt[3]{100 x}}{10}$

Step 5

Verify if $f^{- 1} (x) = \frac{\sqrt[3]{100 x}}{10}$ is the inverse of $f (x) = 10 x^{3}$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $f^{- 1} (10 x^{3})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (10 x^{3}) = \frac{\sqrt[3]{100 (10 x^{3})}}{10}$

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

Multiply $100$ by $10$ .

$f^{- 1} (10 x^{3}) = \frac{\sqrt[3]{1000 x^{3}}}{10}$

Step 5.2.3.2

Rewrite $1000 x^{3}$ as ${(10 x)}^{3}$ .

$f^{- 1} (10 x^{3}) = \frac{\sqrt[3]{{(10 x)}^{3}}}{10}$

Step 5.2.3.3

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

$f^{- 1} (10 x^{3}) = \frac{10 x}{10}$

Step 5.2.4

Cancel the common factor of $10$ .

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Step 5.2.4.1

Cancel the common factor.

$f^{- 1} (10 x^{3}) = \frac{10 x}{10}$

Step 5.2.4.2

Divide $x$ by $1$ .

$f^{- 1} (10 x^{3}) = x$

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $f (\frac{\sqrt[3]{100 x}}{10})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\sqrt[3]{100 x}}{10}) = 10 {(\frac{\sqrt[3]{100 x}}{10})}^{3}$

Step 5.3.3

Apply the product rule to $\frac{\sqrt[3]{100 x}}{10}$ .

$f (\frac{\sqrt[3]{100 x}}{10}) = 10 (\frac{{\sqrt[3]{100 x}}^{3}}{10^{3}})$

Step 5.3.4

Rewrite ${\sqrt[3]{100 x}}^{3}$ as $100 x$ .

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Step 5.3.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{100 x}$ as ${(100 x)}^{\frac{1}{3}}$ .

$f (\frac{\sqrt[3]{100 x}}{10}) = 10 (\frac{{({(100 x)}^{\frac{1}{3}})}^{3}}{10^{3}})$

Step 5.3.4.2

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

$f (\frac{\sqrt[3]{100 x}}{10}) = 10 (\frac{{(100 x)}^{\frac{1}{3} \cdot 3}}{10^{3}})$

Step 5.3.4.3

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

$f (\frac{\sqrt[3]{100 x}}{10}) = 10 (\frac{{(100 x)}^{\frac{3}{3}}}{10^{3}})$

Step 5.3.4.4

Cancel the common factor of $3$ .

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Step 5.3.4.4.1

Cancel the common factor.

$f (\frac{\sqrt[3]{100 x}}{10}) = 10 (\frac{{(100 x)}^{\frac{3}{3}}}{10^{3}})$

Step 5.3.4.4.2

Rewrite the expression.

$f (\frac{\sqrt[3]{100 x}}{10}) = 10 (\frac{100 x}{10^{3}})$

Step 5.3.4.5

Simplify.

$f (\frac{\sqrt[3]{100 x}}{10}) = 10 (\frac{100 x}{10^{3}})$

Step 5.3.5

Raise $10$ to the power of $3$ .

$f (\frac{\sqrt[3]{100 x}}{10}) = 10 (\frac{100 x}{1000})$

Step 5.3.6

Cancel the common factor of $10$ .

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Step 5.3.6.1

Factor $10$ out of $1000$ .

$f (\frac{\sqrt[3]{100 x}}{10}) = 10 (\frac{100 x}{10 (100)})$

Step 5.3.6.2

Cancel the common factor.

$f (\frac{\sqrt[3]{100 x}}{10}) = 10 (\frac{100 x}{10 \cdot 100})$

Step 5.3.6.3

Rewrite the expression.

$f (\frac{\sqrt[3]{100 x}}{10}) = \frac{100 x}{100}$

Step 5.3.7

Cancel the common factor of $100$ .

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Step 5.3.7.1

Cancel the common factor.

$f (\frac{\sqrt[3]{100 x}}{10}) = \frac{100 x}{100}$

Step 5.3.7.2

Divide $x$ by $1$ .

$f (\frac{\sqrt[3]{100 x}}{10}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\sqrt[3]{100 x}}{10}$ is the inverse of $f (x) = 10 x^{3}$ .

$f^{- 1} (x) = \frac{\sqrt[3]{100 x}}{10}$