Finite Math Examples

$f (x) = \frac{1}{x^{5}}$

Step 1

Write $f (x) = \frac{1}{x^{5}}$ as an equation.

$y = \frac{1}{x^{5}}$

Step 2

Interchange the variables.

$x = \frac{1}{y^{5}}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{1}{y^{5}} = x$ .

$\frac{1}{y^{5}} = x$

Step 3.2

Find the LCD of the terms in the equation.

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Step 3.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y^{5}, 1$

Step 3.2.2

The LCM of one and any expression is the expression.

$y^{5}$

Step 3.3

Multiply each term in $\frac{1}{y^{5}} = x$ by $y^{5}$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $\frac{1}{y^{5}} = x$ by $y^{5}$ .

$\frac{1}{y^{5}} y^{5} = x y^{5}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $y^{5}$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{1}{y^{5}} y^{5} = x y^{5}$

Step 3.3.2.1.2

Rewrite the expression.

$1 = x y^{5}$

Step 3.4

Solve the equation.

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Step 3.4.1

Rewrite the equation as $x y^{5} = 1$ .

$x y^{5} = 1$

Step 3.4.2

Divide each term in $x y^{5} = 1$ by $x$ and simplify.

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Step 3.4.2.1

Divide each term in $x y^{5} = 1$ by $x$ .

$\frac{x y^{5}}{x} = \frac{1}{x}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $x$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{x y^{5}}{x} = \frac{1}{x}$

Step 3.4.2.2.1.2

Divide $y^{5}$ by $1$ .

$y^{5} = \frac{1}{x}$

Step 3.4.3

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

$y = \sqrt[5]{\frac{1}{x}}$

Step 3.4.4

Simplify $\sqrt[5]{\frac{1}{x}}$ .

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Step 3.4.4.1

Rewrite $\sqrt[5]{\frac{1}{x}}$ as $\frac{\sqrt[5]{1}}{\sqrt[5]{x}}$ .

$y = \frac{\sqrt[5]{1}}{\sqrt[5]{x}}$

Step 3.4.4.2

Any root of $1$ is $1$ .

$y = \frac{1}{\sqrt[5]{x}}$

Step 3.4.4.3

Multiply $\frac{1}{\sqrt[5]{x}}$ by $\frac{{\sqrt[5]{x}}^{4}}{{\sqrt[5]{x}}^{4}}$ .

$y = \frac{1}{\sqrt[5]{x}} \cdot \frac{{\sqrt[5]{x}}^{4}}{{\sqrt[5]{x}}^{4}}$

Step 3.4.4.4

Combine and simplify the denominator.

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Step 3.4.4.4.1

Multiply $\frac{1}{\sqrt[5]{x}}$ by $\frac{{\sqrt[5]{x}}^{4}}{{\sqrt[5]{x}}^{4}}$ .

$y = \frac{{\sqrt[5]{x}}^{4}}{\sqrt[5]{x} {\sqrt[5]{x}}^{4}}$

Step 3.4.4.4.2

Raise $\sqrt[5]{x}$ to the power of $1$ .

$y = \frac{{\sqrt[5]{x}}^{4}}{{\sqrt[5]{x}}^{1} {\sqrt[5]{x}}^{4}}$

Step 3.4.4.4.3

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

$y = \frac{{\sqrt[5]{x}}^{4}}{{\sqrt[5]{x}}^{1 + 4}}$

Step 3.4.4.4.4

Add $1$ and $4$ .

$y = \frac{{\sqrt[5]{x}}^{4}}{{\sqrt[5]{x}}^{5}}$

Step 3.4.4.4.5

Rewrite ${\sqrt[5]{x}}^{5}$ as $x$ .

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Step 3.4.4.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x}$ as $x^{\frac{1}{5}}$ .

$y = \frac{{\sqrt[5]{x}}^{4}}{{(x^{\frac{1}{5}})}^{5}}$

Step 3.4.4.4.5.2

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

$y = \frac{{\sqrt[5]{x}}^{4}}{x^{\frac{1}{5} \cdot 5}}$

Step 3.4.4.4.5.3

Combine $\frac{1}{5}$ and $5$ .

$y = \frac{{\sqrt[5]{x}}^{4}}{x^{\frac{5}{5}}}$

Step 3.4.4.4.5.4

Cancel the common factor of $5$ .

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Step 3.4.4.4.5.4.1

Cancel the common factor.

$y = \frac{{\sqrt[5]{x}}^{4}}{x^{\frac{5}{5}}}$

Step 3.4.4.4.5.4.2

Rewrite the expression.

$y = \frac{{\sqrt[5]{x}}^{4}}{x^{1}}$

Step 3.4.4.4.5.5

Simplify.

$y = \frac{{\sqrt[5]{x}}^{4}}{x}$

Step 3.4.4.5

Rewrite ${\sqrt[5]{x}}^{4}$ as $\sqrt[5]{x^{4}}$ .

$y = \frac{\sqrt[5]{x^{4}}}{x}$

Step 4

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

$f^{- 1} (x) = \frac{\sqrt[5]{x^{4}}}{x}$

Step 5

Verify if $f^{- 1} (x) = \frac{\sqrt[5]{x^{4}}}{x}$ is the inverse of $f (x) = \frac{1}{x^{5}}$ .

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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 $f^{- 1} (\frac{1}{x^{5}})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{1}{x^{5}}) = \frac{\sqrt[5]{{(\frac{1}{x^{5}})}^{4}}}{\frac{1}{x^{5}}}$

Step 5.2.3

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{1}{x^{5}}) = \sqrt[5]{{(\frac{1}{x^{5}})}^{4}} x^{5}$

Step 5.2.4

Apply the product rule to $\frac{1}{x^{5}}$ .

$f^{- 1} (\frac{1}{x^{5}}) = \sqrt[5]{\frac{1^{4}}{{(x^{5})}^{4}}} x^{5}$

Step 5.2.5

One to any power is one.

$f^{- 1} (\frac{1}{x^{5}}) = \sqrt[5]{\frac{1}{{(x^{5})}^{4}}} x^{5}$

Step 5.2.6

Multiply the exponents in ${(x^{5})}^{4}$ .

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Step 5.2.6.1

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

$f^{- 1} (\frac{1}{x^{5}}) = \sqrt[5]{\frac{1}{x^{5 \cdot 4}}} x^{5}$

Step 5.2.6.2

Multiply $5$ by $4$ .

$f^{- 1} (\frac{1}{x^{5}}) = \sqrt[5]{\frac{1}{x^{20}}} x^{5}$

Step 5.2.7

Rewrite $1$ as $1^{5}$ .

$f^{- 1} (\frac{1}{x^{5}}) = \sqrt[5]{\frac{1^{5}}{x^{20}}} x^{5}$

Step 5.2.8

Rewrite $x^{20}$ as ${(x^{4})}^{5}$ .

$f^{- 1} (\frac{1}{x^{5}}) = \sqrt[5]{\frac{1^{5}}{{(x^{4})}^{5}}} x^{5}$

Step 5.2.9

Rewrite $\frac{1^{5}}{{(x^{4})}^{5}}$ as ${(\frac{1}{x^{4}})}^{5}$ .

$f^{- 1} (\frac{1}{x^{5}}) = \sqrt[5]{{(\frac{1}{x^{4}})}^{5}} x^{5}$

Step 5.2.10

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

$f^{- 1} (\frac{1}{x^{5}}) = \frac{1}{x^{4}} \cdot x^{5}$

Step 5.2.11

Cancel the common factor of $x^{4}$ .

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Step 5.2.11.1

Factor $x^{4}$ out of $x^{5}$ .

$f^{- 1} (\frac{1}{x^{5}}) = \frac{1}{x^{4}} \cdot (x^{4} x)$

Step 5.2.11.2

Cancel the common factor.

$f^{- 1} (\frac{1}{x^{5}}) = \frac{1}{x^{4}} \cdot (x^{4} x)$

Step 5.2.11.3

Rewrite the expression.

$f^{- 1} (\frac{1}{x^{5}}) = x$

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 $f (\frac{\sqrt[5]{x^{4}}}{x})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\sqrt[5]{x^{4}}}{x}) = \frac{1}{{(\frac{\sqrt[5]{x^{4}}}{x})}^{5}}$

Step 5.3.3

Simplify the denominator.

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Step 5.3.3.1

Apply the product rule to $\frac{\sqrt[5]{x^{4}}}{x}$ .

$f (\frac{\sqrt[5]{x^{4}}}{x}) = \frac{1}{\frac{{\sqrt[5]{x^{4}}}^{5}}{x^{5}}}$

Step 5.3.3.2

Rewrite ${\sqrt[5]{x^{4}}}^{5}$ as $x^{4}$ .

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Step 5.3.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x^{4}}$ as $x^{\frac{4}{5}}$ .

$f (\frac{\sqrt[5]{x^{4}}}{x}) = \frac{1}{\frac{{(x^{\frac{4}{5}})}^{5}}{x^{5}}}$

Step 5.3.3.2.2

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

$f (\frac{\sqrt[5]{x^{4}}}{x}) = \frac{1}{\frac{x^{\frac{4}{5} \cdot 5}}{x^{5}}}$

Step 5.3.3.2.3

Combine $\frac{4}{5}$ and $5$ .

$f (\frac{\sqrt[5]{x^{4}}}{x}) = \frac{1}{\frac{x^{\frac{4 \cdot 5}{5}}}{x^{5}}}$

Step 5.3.3.2.4

Multiply $4$ by $5$ .

$f (\frac{\sqrt[5]{x^{4}}}{x}) = \frac{1}{\frac{x^{\frac{20}{5}}}{x^{5}}}$

Step 5.3.3.2.5

Cancel the common factor of $20$ and $5$ .

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Step 5.3.3.2.5.1

Factor $5$ out of $20$ .

$f (\frac{\sqrt[5]{x^{4}}}{x}) = \frac{1}{\frac{x^{\frac{5 \cdot 4}{5}}}{x^{5}}}$

Step 5.3.3.2.5.2

Cancel the common factors.

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Step 5.3.3.2.5.2.1

Factor $5$ out of $5$ .

$f (\frac{\sqrt[5]{x^{4}}}{x}) = \frac{1}{\frac{x^{\frac{5 \cdot 4}{5 (1)}}}{x^{5}}}$

Step 5.3.3.2.5.2.2

Cancel the common factor.

$f (\frac{\sqrt[5]{x^{4}}}{x}) = \frac{1}{\frac{x^{\frac{5 \cdot 4}{5 \cdot 1}}}{x^{5}}}$

Step 5.3.3.2.5.2.3

Rewrite the expression.

$f (\frac{\sqrt[5]{x^{4}}}{x}) = \frac{1}{\frac{x^{\frac{4}{1}}}{x^{5}}}$

Step 5.3.3.2.5.2.4

Divide $4$ by $1$ .

$f (\frac{\sqrt[5]{x^{4}}}{x}) = \frac{1}{\frac{x^{4}}{x^{5}}}$

Step 5.3.3.3

Cancel the common factor of $x^{4}$ and $x^{5}$ .

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Step 5.3.3.3.1

Multiply by $1$ .

$f (\frac{\sqrt[5]{x^{4}}}{x}) = \frac{1}{\frac{x^{4} \cdot 1}{x^{5}}}$

Step 5.3.3.3.2

Cancel the common factors.

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Step 5.3.3.3.2.1

Factor $x^{4}$ out of $x^{5}$ .

$f (\frac{\sqrt[5]{x^{4}}}{x}) = \frac{1}{\frac{x^{4} \cdot 1}{x^{4} x}}$

Step 5.3.3.3.2.2

Cancel the common factor.

$f (\frac{\sqrt[5]{x^{4}}}{x}) = \frac{1}{\frac{x^{4} \cdot 1}{x^{4} x}}$

Step 5.3.3.3.2.3

Rewrite the expression.

$f (\frac{\sqrt[5]{x^{4}}}{x}) = \frac{1}{\frac{1}{x}}$

Step 5.3.4

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{\sqrt[5]{x^{4}}}{x}) = 1 x$

Step 5.3.5

Multiply $x$ by $1$ .

$f (\frac{\sqrt[5]{x^{4}}}{x}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\sqrt[5]{x^{4}}}{x}$ is the inverse of $f (x) = \frac{1}{x^{5}}$ .

$f^{- 1} (x) = \frac{\sqrt[5]{x^{4}}}{x}$