Algebra Examples

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply the equation by $y - 1$ .

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

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.

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

Step 3.2.1.2

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

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

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.

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

Step 3.3.1.2

Rewrite the expression.

$y x - x = y + 1$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Subtract $y$ from both sides of the equation.

$y x - x - y = 1$

Step 3.4.2

Add $x$ to both sides of the equation.

$y x - y = 1 + x$

Step 3.4.3

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

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Step 3.4.3.1

Factor $y$ out of $y x$ .

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

Step 3.4.3.2

Factor $y$ out of $- y$ .

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

Step 3.4.3.3

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

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

Step 3.4.4

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

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Step 3.4.4.1

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

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

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.

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

Step 3.4.4.2.1.2

Divide $y$ by $1$ .

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

Step 3.4.4.3

Simplify the right side.

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Step 3.4.4.3.1

Combine the numerators over the common denominator.

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

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Remove parentheses.

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

Step 5.2.4

Multiply the numerator and denominator of the fraction by $x - 1$ .

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Step 5.2.4.1

Multiply $\frac{1 + \frac{x + 1}{x - 1}}{\frac{x + 1}{x - 1} - 1}$ by $\frac{x - 1}{x - 1}$ .

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

Step 5.2.4.2

Combine.

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

Step 5.2.5

Apply the distributive property.

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

Step 5.2.6

Simplify by cancelling.

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Step 5.2.6.1

Cancel the common factor of $x - 1$ .

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Step 5.2.6.1.1

Cancel the common factor.

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

Step 5.2.6.1.2

Rewrite the expression.

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

Step 5.2.6.2

Cancel the common factor of $x - 1$ .

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Step 5.2.6.2.1

Cancel the common factor.

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

Step 5.2.6.2.2

Rewrite the expression.

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

Step 5.2.7

Simplify the numerator.

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Step 5.2.7.1

Multiply $x - 1$ by $1$ .

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

Step 5.2.7.2

Add $x$ and $x$ .

$f^{- 1} (\frac{x + 1}{x - 1}) = \frac{2 x - 1 + 1}{x + 1 + (x - 1) \cdot - 1}$

Step 5.2.7.3

Add $- 1$ and $1$ .

$f^{- 1} (\frac{x + 1}{x - 1}) = \frac{2 x + 0}{x + 1 + (x - 1) \cdot - 1}$

Step 5.2.7.4

Add $2 x$ and $0$ .

$f^{- 1} (\frac{x + 1}{x - 1}) = \frac{2 x}{x + 1 + (x - 1) \cdot - 1}$

Step 5.2.8

Simplify the denominator.

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Step 5.2.8.1

Apply the distributive property.

$f^{- 1} (\frac{x + 1}{x - 1}) = \frac{2 x}{x + 1 + x \cdot - 1 - 1 \cdot - 1}$

Step 5.2.8.2

Move $- 1$ to the left of $x$ .

$f^{- 1} (\frac{x + 1}{x - 1}) = \frac{2 x}{x + 1 - 1 \cdot x - 1 \cdot - 1}$

Step 5.2.8.3

Multiply $- 1$ by $- 1$ .

$f^{- 1} (\frac{x + 1}{x - 1}) = \frac{2 x}{x + 1 - 1 \cdot x + 1}$

Step 5.2.8.4

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

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

Step 5.2.8.5

Subtract $x$ from $x$ .

$f^{- 1} (\frac{x + 1}{x - 1}) = \frac{2 x}{0 + 1 + 1}$

Step 5.2.8.6

Add $0$ and $1$ .

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

Step 5.2.8.7

Add $1$ and $1$ .

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

Step 5.2.9

Cancel the common factor of $2$ .

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Step 5.2.9.1

Cancel the common factor.

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

Step 5.2.9.2

Divide $x$ by $1$ .

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

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

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

Step 5.3.3

Multiply the numerator and denominator of the fraction by $x - 1$ .

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Step 5.3.3.1

Multiply $\frac{\frac{1 + x}{x - 1} + 1}{\frac{1 + x}{x - 1} - 1}$ by $\frac{x - 1}{x - 1}$ .

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

Step 5.3.3.2

Combine.

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

Step 5.3.4

Apply the distributive property.

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

Step 5.3.5

Simplify by cancelling.

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Step 5.3.5.1

Cancel the common factor of $x - 1$ .

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Step 5.3.5.1.1

Cancel the common factor.

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

Step 5.3.5.1.2

Rewrite the expression.

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

Step 5.3.5.2

Cancel the common factor of $x - 1$ .

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Step 5.3.5.2.1

Cancel the common factor.

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

Step 5.3.5.2.2

Rewrite the expression.

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

Step 5.3.6

Simplify the numerator.

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Step 5.3.6.1

Multiply $x - 1$ by $1$ .

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

Step 5.3.6.2

Subtract $1$ from $1$ .

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

Step 5.3.6.3

Add $x + x$ and $0$ .

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

Step 5.3.6.4

Add $x$ and $x$ .

$f (\frac{1 + x}{x - 1}) = \frac{2 x}{1 + x + (x - 1) \cdot - 1}$

Step 5.3.7

Simplify the denominator.

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Step 5.3.7.1

Apply the distributive property.

$f (\frac{1 + x}{x - 1}) = \frac{2 x}{1 + x + x \cdot - 1 - 1 \cdot - 1}$

Step 5.3.7.2

Move $- 1$ to the left of $x$ .

$f (\frac{1 + x}{x - 1}) = \frac{2 x}{1 + x - 1 \cdot x - 1 \cdot - 1}$

Step 5.3.7.3

Multiply $- 1$ by $- 1$ .

$f (\frac{1 + x}{x - 1}) = \frac{2 x}{1 + x - 1 \cdot x + 1}$

Step 5.3.7.4

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

$f (\frac{1 + x}{x - 1}) = \frac{2 x}{1 + x - x + 1}$

Step 5.3.7.5

Add $1$ and $1$ .

$f (\frac{1 + x}{x - 1}) = \frac{2 x}{x - x + 2}$

Step 5.3.7.6

Subtract $x$ from $x$ .

$f (\frac{1 + x}{x - 1}) = \frac{2 x}{0 + 2}$

Step 5.3.7.7

Add $0$ and $2$ .

$f (\frac{1 + x}{x - 1}) = \frac{2 x}{2}$

Step 5.3.8

Cancel the common factor of $2$ .

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Step 5.3.8.1

Cancel the common factor.

$f (\frac{1 + x}{x - 1}) = \frac{2 x}{2}$

Step 5.3.8.2

Divide $x$ by $1$ .

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

Step 5.4

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

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