Calculus Examples

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply the equation by $y - 1$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify $(y - 1) (x)$ .

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Step 3.2.1.1

Apply the distributive property.

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

Step 3.2.1.2

Rewrite $- 1 x$ as $- x$ .

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $y - 1$ .

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Step 3.3.1.1

Cancel the common factor.

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

Step 3.3.1.2

Rewrite the expression.

$y x - x = y$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Subtract $y$ from both sides of the equation.

$y x - x - y = 0$

Step 3.4.2

Add $x$ to both sides of the equation.

$y x - y = x$

Step 3.4.3

Factor $y$ out of $y x - y$ .

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Step 3.4.3.1

Factor $y$ out of $y x$ .

$y (x) - y = x$

Step 3.4.3.2

Factor $y$ out of $- y$ .

$y (x) + y \cdot - 1 = x$

Step 3.4.3.3

Factor $y$ out of $y (x) + y \cdot - 1$ .

$y (x - 1) = x$

Step 3.4.4

Divide each term in $y (x - 1) = x$ by $x - 1$ and simplify.

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Step 3.4.4.1

Divide each term in $y (x - 1) = x$ by $x - 1$ .

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

Step 3.4.4.2

Simplify the left side.

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Step 3.4.4.2.1

Cancel the common factor of $x - 1$ .

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Step 3.4.4.2.1.1

Cancel the common factor.

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

Step 3.4.4.2.1.2

Divide $y$ by $1$ .

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

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.2.4

Simplify the denominator.

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Step 5.2.4.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

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

Step 5.2.4.2

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

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

Step 5.2.4.3

Combine the numerators over the common denominator.

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

Step 5.2.4.4

Rewrite $\frac{x - (x - 1)}{x - 1}$ in a factored form.

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Step 5.2.4.4.1

Apply the distributive property.

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

Step 5.2.4.4.2

Multiply $- 1$ by $- 1$ .

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

Step 5.2.4.4.3

Subtract $x$ from $x$ .

$f^{- 1} (\frac{x}{x - 1}) = \frac{x}{x - 1} \cdot \frac{1}{\frac{0 + 1}{x - 1}}$

Step 5.2.4.4.4

Add $0$ and $1$ .

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

Step 5.2.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.2.6

Cancel the common factor of $x - 1$ .

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Step 5.2.6.1

Factor $x - 1$ out of $1 (x - 1)$ .

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

Step 5.2.6.2

Cancel the common factor.

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

Step 5.2.6.3

Rewrite the expression.

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

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

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

Step 5.3.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.3.4

Simplify the denominator.

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Step 5.3.4.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

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

Step 5.3.4.2

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

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

Step 5.3.4.3

Combine the numerators over the common denominator.

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

Step 5.3.4.4

Rewrite $\frac{x - (x - 1)}{x - 1}$ in a factored form.

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Step 5.3.4.4.1

Apply the distributive property.

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

Step 5.3.4.4.2

Multiply $- 1$ by $- 1$ .

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

Step 5.3.4.4.3

Subtract $x$ from $x$ .

$f (\frac{x}{x - 1}) = \frac{x}{x - 1} \cdot \frac{1}{\frac{0 + 1}{x - 1}}$

Step 5.3.4.4.4

Add $0$ and $1$ .

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

Step 5.3.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.3.6

Cancel the common factor of $x - 1$ .

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Step 5.3.6.1

Factor $x - 1$ out of $1 (x - 1)$ .

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

Step 5.3.6.2

Cancel the common factor.

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

Step 5.3.6.3

Rewrite the expression.

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

Step 5.4

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

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