Algebra Examples

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

Step 1

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

$y = \frac{1}{f (x)}$

Step 2

Interchange the variables.

$f = \frac{1}{y (x)}$

Step 3

Solve for $y$ .

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Step 3.1

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

$\frac{1}{y x} = f$

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.

$y x, 1$

Step 3.2.2

The LCM of one and any expression is the expression.

$y x$

Step 3.3

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

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Step 3.3.1

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

$\frac{1}{y x} (y x) = f (y x)$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $y x$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{1}{y x} (y x) = f (y x)$

Step 3.3.2.1.2

Rewrite the expression.

$1 = f (y x)$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Remove parentheses.

$1 = f y x$

Step 3.4

Solve the equation.

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Step 3.4.1

Rewrite the equation as $f y x = 1$ .

$f y x = 1$

Step 3.4.2

Divide each term in $f y x = 1$ by $f x$ and simplify.

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Step 3.4.2.1

Divide each term in $f y x = 1$ by $f x$ .

$\frac{f y x}{f x} = \frac{1}{f x}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $f$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{f y x}{f x} = \frac{1}{f x}$

Step 3.4.2.2.1.2

Rewrite the expression.

$\frac{y x}{x} = \frac{1}{f x}$

Step 3.4.2.2.2

Cancel the common factor of $x$ .

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Step 3.4.2.2.2.1

Cancel the common factor.

$\frac{y x}{x} = \frac{1}{f x}$

Step 3.4.2.2.2.2

Divide $y$ by $1$ .

$y = \frac{1}{f x}$

Step 4

Replace $y$ with $f (f)$ to show the final answer.

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

Step 5

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

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

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

$y = \frac{1}{f x}$

Step 5.2.2

Interchange the variables.

$f = \frac{1}{y x}$

Step 5.2.3

Solve for $y$ .

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Step 5.2.3.1

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

$\frac{1}{y x} = f$

Step 5.2.3.2

Find the LCD of the terms in the equation.

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Step 5.2.3.2.1

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

$y x, 1$

Step 5.2.3.2.2

The LCM of one and any expression is the expression.

$y x$

Step 5.2.3.3

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

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Step 5.2.3.3.1

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

$\frac{1}{y x} (y x) = f (y x)$

Step 5.2.3.3.2

Simplify the left side.

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Step 5.2.3.3.2.1

Cancel the common factor of $y x$ .

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Step 5.2.3.3.2.1.1

Cancel the common factor.

$\frac{1}{y x} (y x) = f (y x)$

Step 5.2.3.3.2.1.2

Rewrite the expression.

$1 = f (y x)$

Step 5.2.3.3.3

Simplify the right side.

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Step 5.2.3.3.3.1

Remove parentheses.

$1 = f y x$

Step 5.2.3.4

Solve the equation.

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Step 5.2.3.4.1

Rewrite the equation as $f y x = 1$ .

$f y x = 1$

Step 5.2.3.4.2

Divide each term in $f y x = 1$ by $f x$ and simplify.

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Step 5.2.3.4.2.1

Divide each term in $f y x = 1$ by $f x$ .

$\frac{f y x}{f x} = \frac{1}{f x}$

Step 5.2.3.4.2.2

Simplify the left side.

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Step 5.2.3.4.2.2.1

Cancel the common factor of $f$ .

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Step 5.2.3.4.2.2.1.1

Cancel the common factor.

$\frac{f y x}{f x} = \frac{1}{f x}$

Step 5.2.3.4.2.2.1.2

Rewrite the expression.

$\frac{y x}{x} = \frac{1}{f x}$

Step 5.2.3.4.2.2.2

Cancel the common factor of $x$ .

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Step 5.2.3.4.2.2.2.1

Cancel the common factor.

$\frac{y x}{x} = \frac{1}{f x}$

Step 5.2.3.4.2.2.2.2

Divide $y$ by $1$ .

$y = \frac{1}{f x}$

Step 5.2.4

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

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

Step 5.2.5

Set up the composite result function.

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

Step 5.2.6

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

$f (\frac{1}{f x}) = \frac{1}{(\frac{1}{f x}) \cdot x}$

Step 5.2.7

Combine $\frac{1}{f x}$ and $x$ .

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

Step 5.2.8

Reduce the expression $\frac{x}{f x}$ by cancelling the common factors.

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Step 5.2.8.1

Cancel the common factor.

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

Step 5.2.8.2

Rewrite the expression.

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

Step 5.2.9

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.2.10

Multiply $f$ by $1$ .

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

Step 5.3

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

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Step 5.3.1

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

$y = \frac{1}{f x}$

Step 5.3.2

Interchange the variables.

$f = \frac{1}{y x}$

Step 5.3.3

Solve for $y$ .

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Step 5.3.3.1

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

$\frac{1}{y x} = f$

Step 5.3.3.2

Find the LCD of the terms in the equation.

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Step 5.3.3.2.1

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

$y x, 1$

Step 5.3.3.2.2

The LCM of one and any expression is the expression.

$y x$

Step 5.3.3.3

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

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Step 5.3.3.3.1

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

$\frac{1}{y x} (y x) = f (y x)$

Step 5.3.3.3.2

Simplify the left side.

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Step 5.3.3.3.2.1

Cancel the common factor of $y x$ .

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Step 5.3.3.3.2.1.1

Cancel the common factor.

$\frac{1}{y x} (y x) = f (y x)$

Step 5.3.3.3.2.1.2

Rewrite the expression.

$1 = f (y x)$

Step 5.3.3.3.3

Simplify the right side.

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Step 5.3.3.3.3.1

Remove parentheses.

$1 = f y x$

Step 5.3.3.4

Solve the equation.

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Step 5.3.3.4.1

Rewrite the equation as $f y x = 1$ .

$f y x = 1$

Step 5.3.3.4.2

Divide each term in $f y x = 1$ by $f x$ and simplify.

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Step 5.3.3.4.2.1

Divide each term in $f y x = 1$ by $f x$ .

$\frac{f y x}{f x} = \frac{1}{f x}$

Step 5.3.3.4.2.2

Simplify the left side.

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Step 5.3.3.4.2.2.1

Cancel the common factor of $f$ .

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Step 5.3.3.4.2.2.1.1

Cancel the common factor.

$\frac{f y x}{f x} = \frac{1}{f x}$

Step 5.3.3.4.2.2.1.2

Rewrite the expression.

$\frac{y x}{x} = \frac{1}{f x}$

Step 5.3.3.4.2.2.2

Cancel the common factor of $x$ .

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Step 5.3.3.4.2.2.2.1

Cancel the common factor.

$\frac{y x}{x} = \frac{1}{f x}$

Step 5.3.3.4.2.2.2.2

Divide $y$ by $1$ .

$y = \frac{1}{f x}$

Step 5.3.4

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

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

Step 5.3.5

Set up the composite result function.

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

Step 5.3.6

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

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

Step 5.3.7

Combine $\frac{1}{f x}$ and $x$ .

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

Step 5.3.8

Reduce the expression $\frac{x}{f x}$ by cancelling the common factors.

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Step 5.3.8.1

Cancel the common factor.

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

Step 5.3.8.2

Rewrite the expression.

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

Step 5.3.9

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.3.10

Multiply $f$ by $1$ .

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

Step 5.4

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

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