Find the Inverse (x+5)/(x-6)

$\frac{x + 5}{x - 6}$

Step 1

Interchange the variables.

$x = \frac{y + 5}{y - 6}$

Step 2

Solve for $y$ .

Multiply the equation by $y - 6$ .

$(y - 6) (x) = (\frac{y + 5}{y - 6}) (y - 6)$

Simplify the left side.

Apply the distributive property.

$y x - 6 x = (\frac{y + 5}{y - 6}) (y - 6)$

Simplify the right side.

Cancel the common factor of $y - 6$ .

Cancel the common factor.

$y x - 6 x = \frac{y + 5}{y - 6} (y - 6)$

Rewrite the expression.

$y x - 6 x = y + 5$

Solve for $y$ .

Subtract $y$ from both sides of the equation.

$y x - 6 x - y = 5$

Add $6 x$ to both sides of the equation.

$y x - y = 5 + 6 x$

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

Factor $y$ out of $y x$ .

$y (x) - y = 5 + 6 x$

Factor $y$ out of $- y$ .

$y (x) + y \cdot - 1 = 5 + 6 x$

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

$y (x - 1) = 5 + 6 x$

Divide each term in $y (x - 1) = 5 + 6 x$ by $x - 1$ and simplify.

Divide each term in $y (x - 1) = 5 + 6 x$ by $x - 1$ .

$\frac{y (x - 1)}{x - 1} = \frac{5}{x - 1} + \frac{6 x}{x - 1}$

Simplify the left side.

Cancel the common factor of $x - 1$ .

Cancel the common factor.

$\frac{y (x - 1)}{x - 1} = \frac{5}{x - 1} + \frac{6 x}{x - 1}$

Divide $y$ by $1$ .

$y = \frac{5}{x - 1} + \frac{6 x}{x - 1}$

Simplify the right side.

Combine the numerators over the common denominator.

$y = \frac{5 + 6 x}{x - 1}$

Step 3

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

$f^{- 1} (x) = \frac{5 + 6 x}{x - 1}$

Step 4

Verify if $f^{- 1} (x) = \frac{5 + 6 x}{x - 1}$ is the inverse of $f (x) = \frac{x + 5}{x - 6}$ .

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

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

Set up the composite result function.

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

Evaluate $f^{- 1} (\frac{x + 5}{x - 6})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{x + 5}{x - 6}) = \frac{5 + 6 (\frac{x + 5}{x - 6})}{(\frac{x + 5}{x - 6}) - 1}$

Multiply the numerator and denominator of the complex fraction by $x - 6$ .

Multiply $\frac{5 + 6 \frac{x + 5}{x - 6}}{\frac{x + 5}{x - 6} - 1}$ by $\frac{x - 6}{x - 6}$ .

$f^{- 1} (\frac{x + 5}{x - 6}) = \frac{x - 6}{x - 6} \cdot \frac{5 + 6 (\frac{x + 5}{x - 6})}{\frac{x + 5}{x - 6} - 1}$

Combine.

$f^{- 1} (\frac{x + 5}{x - 6}) = \frac{(x - 6) (5 + 6 (\frac{x + 5}{x - 6}))}{(x - 6) (\frac{x + 5}{x - 6} - 1)}$

Apply the distributive property.

$f^{- 1} (\frac{x + 5}{x - 6}) = \frac{(x - 6) \cdot 5 + (x - 6) (6 (\frac{x + 5}{x - 6}))}{(x - 6) (\frac{x + 5}{x - 6}) + (x - 6) \cdot - 1}$

Cancel the common factor of $x - 6$ .

Cancel the common factor.

$f^{- 1} (\frac{x + 5}{x - 6}) = \frac{(x - 6) \cdot 5 + (x - 6) (6 (\frac{x + 5}{x - 6}))}{(x - 6) (\frac{x + 5}{x - 6}) + (x - 6) \cdot - 1}$

Rewrite the expression.

$f^{- 1} (\frac{x + 5}{x - 6}) = \frac{(x - 6) \cdot 5 + (x - 6) (6 (\frac{x + 5}{x - 6}))}{x + 5 + (x - 6) \cdot - 1}$

Simplify the numerator.

Factor $x - 6$ out of $(x - 6) \cdot 5 + (x - 6) \cdot 6 \frac{x + 5}{x - 6}$ .

Factor $x - 6$ out of $(x - 6) \cdot 5$ .

$f^{- 1} (\frac{x + 5}{x - 6}) = \frac{(x - 6) (5) + (x - 6) (6 (\frac{x + 5}{x - 6}))}{x + 5 + (x - 6) \cdot - 1}$

Factor $x - 6$ out of $(x - 6) (5) + (x - 6) (6 \frac{x + 5}{x - 6})$ .

$f^{- 1} (\frac{x + 5}{x - 6}) = \frac{(x - 6) (5 + 6 (\frac{x + 5}{x - 6}))}{x + 5 + (x - 6) \cdot - 1}$

Combine $6$ and $\frac{x + 5}{x - 6}$ .

$f^{- 1} (\frac{x + 5}{x - 6}) = \frac{(x - 6) (5 + \frac{6 (x + 5)}{x - 6})}{x + 5 + (x - 6) \cdot - 1}$

To write $5$ as a fraction with a common denominator, multiply by $\frac{x - 6}{x - 6}$ .

$f^{- 1} (\frac{x + 5}{x - 6}) = \frac{(x - 6) (\frac{5 (x - 6)}{x - 6} + \frac{6 (x + 5)}{x - 6})}{x + 5 + (x - 6) \cdot - 1}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{x + 5}{x - 6}) = \frac{(x - 6) (\frac{5 (x - 6) + 6 (x + 5)}{x - 6})}{x + 5 + (x - 6) \cdot - 1}$

Reorder terms.

$f^{- 1} (\frac{x + 5}{x - 6}) = \frac{(x - 6) (\frac{6 (x + 5) + 5 (x - 6)}{x - 6})}{x + 5 + (x - 6) \cdot - 1}$

Rewrite $\frac{6 (x + 5) + 5 (x - 6)}{x - 6}$ in a factored form.

Apply the distributive property.

$f^{- 1} (\frac{x + 5}{x - 6}) = \frac{(x - 6) (\frac{6 x + 6 \cdot 5 + 5 (x - 6)}{x - 6})}{x + 5 + (x - 6) \cdot - 1}$

Multiply $6$ by $5$ .

$f^{- 1} (\frac{x + 5}{x - 6}) = \frac{(x - 6) (\frac{6 x + 30 + 5 (x - 6)}{x - 6})}{x + 5 + (x - 6) \cdot - 1}$

Apply the distributive property.

$f^{- 1} (\frac{x + 5}{x - 6}) = \frac{(x - 6) (\frac{6 x + 30 + 5 x + 5 \cdot - 6}{x - 6})}{x + 5 + (x - 6) \cdot - 1}$

Multiply $5$ by $- 6$ .

$f^{- 1} (\frac{x + 5}{x - 6}) = \frac{(x - 6) (\frac{6 x + 30 + 5 x - 30}{x - 6})}{x + 5 + (x - 6) \cdot - 1}$

Add $6 x$ and $5 x$ .

$f^{- 1} (\frac{x + 5}{x - 6}) = \frac{(x - 6) (\frac{11 x + 30 - 30}{x - 6})}{x + 5 + (x - 6) \cdot - 1}$

Subtract $30$ from $30$ .

$f^{- 1} (\frac{x + 5}{x - 6}) = \frac{(x - 6) (\frac{11 x + 0}{x - 6})}{x + 5 + (x - 6) \cdot - 1}$

Add $11 x$ and $0$ .

$f^{- 1} (\frac{x + 5}{x - 6}) = \frac{(x - 6) (\frac{11 x}{x - 6})}{x + 5 + (x - 6) \cdot - 1}$

Simplify the denominator.

Apply the distributive property.

$f^{- 1} (\frac{x + 5}{x - 6}) = \frac{(x - 6) (\frac{11 x}{x - 6})}{x + 5 + x \cdot - 1 - 6 \cdot - 1}$

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

$f^{- 1} (\frac{x + 5}{x - 6}) = \frac{(x - 6) (\frac{11 x}{x - 6})}{x + 5 - 1 \cdot x - 6 \cdot - 1}$

Multiply $- 6$ by $- 1$ .

$f^{- 1} (\frac{x + 5}{x - 6}) = \frac{(x - 6) (\frac{11 x}{x - 6})}{x + 5 - 1 \cdot x + 6}$

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

$f^{- 1} (\frac{x + 5}{x - 6}) = \frac{(x - 6) (\frac{11 x}{x - 6})}{x + 5 - x + 6}$

Subtract $x$ from $x$ .

$f^{- 1} (\frac{x + 5}{x - 6}) = \frac{(x - 6) (\frac{11 x}{x - 6})}{0 + 5 + 6}$

Add $0$ and $5$ .

$f^{- 1} (\frac{x + 5}{x - 6}) = \frac{(x - 6) (\frac{11 x}{x - 6})}{5 + 6}$

Add $5$ and $6$ .

$f^{- 1} (\frac{x + 5}{x - 6}) = \frac{(x - 6) (\frac{11 x}{x - 6})}{11}$

Reduce the expression by cancelling the common factors.

Factor $\frac{11 x}{x - 6}$ out of $\frac{(x - 6) \frac{11 x}{x - 6}}{11}$ .

$f^{- 1} (\frac{x + 5}{x - 6}) = \frac{11 x}{x - 6} \cdot \frac{x - 6}{11}$

Cancel the common factor of $11$ .

Factor $11$ out of $11 x$ .

$f^{- 1} (\frac{x + 5}{x - 6}) = \frac{11 (x)}{x - 6} \cdot \frac{x - 6}{11}$

Cancel the common factor.

$f^{- 1} (\frac{x + 5}{x - 6}) = \frac{11 x}{x - 6} \cdot \frac{x - 6}{11}$

Rewrite the expression.

$f^{- 1} (\frac{x + 5}{x - 6}) = \frac{x}{x - 6} \cdot (x - 6)$

Cancel the common factor of $x - 6$ .

Cancel the common factor.

$f^{- 1} (\frac{x + 5}{x - 6}) = \frac{x}{x - 6} \cdot (x - 6)$

Rewrite the expression.

$f^{- 1} (\frac{x + 5}{x - 6}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{5 + 6 x}{x - 1})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{5 + 6 x}{x - 1}) = \frac{(\frac{5 + 6 x}{x - 1}) + 5}{(\frac{5 + 6 x}{x - 1}) - 6}$

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

Multiply $\frac{\frac{5 + 6 x}{x - 1} + 5}{\frac{5 + 6 x}{x - 1} - 6}$ by $\frac{x - 1}{x - 1}$ .

$f (\frac{5 + 6 x}{x - 1}) = \frac{x - 1}{x - 1} \cdot \frac{\frac{5 + 6 x}{x - 1} + 5}{\frac{5 + 6 x}{x - 1} - 6}$

Combine.

$f (\frac{5 + 6 x}{x - 1}) = \frac{(x - 1) (\frac{5 + 6 x}{x - 1} + 5)}{(x - 1) (\frac{5 + 6 x}{x - 1} - 6)}$

Apply the distributive property.

$f (\frac{5 + 6 x}{x - 1}) = \frac{(x - 1) (\frac{5 + 6 x}{x - 1}) + (x - 1) \cdot 5}{(x - 1) (\frac{5 + 6 x}{x - 1}) + (x - 1) \cdot - 6}$

Simplify by cancelling.

Cancel the common factor of $x - 1$ .

Cancel the common factor.

$f (\frac{5 + 6 x}{x - 1}) = \frac{(x - 1) (\frac{5 + 6 x}{x - 1}) + (x - 1) \cdot 5}{(x - 1) (\frac{5 + 6 x}{x - 1}) + (x - 1) \cdot - 6}$

Rewrite the expression.

$f (\frac{5 + 6 x}{x - 1}) = \frac{5 + 6 x + (x - 1) \cdot 5}{(x - 1) (\frac{5 + 6 x}{x - 1}) + (x - 1) \cdot - 6}$

Cancel the common factor of $x - 1$ .

Cancel the common factor.

$f (\frac{5 + 6 x}{x - 1}) = \frac{5 + 6 x + (x - 1) \cdot 5}{(x - 1) (\frac{5 + 6 x}{x - 1}) + (x - 1) \cdot - 6}$

Rewrite the expression.

$f (\frac{5 + 6 x}{x - 1}) = \frac{5 + 6 x + (x - 1) \cdot 5}{5 + 6 x + (x - 1) \cdot - 6}$

Simplify the numerator.

Apply the distributive property.

$f (\frac{5 + 6 x}{x - 1}) = \frac{5 + 6 x + x \cdot 5 - 1 \cdot 5}{5 + 6 x + (x - 1) \cdot - 6}$

Move $5$ to the left of $x$ .

$f (\frac{5 + 6 x}{x - 1}) = \frac{5 + 6 x + 5 \cdot x - 1 \cdot 5}{5 + 6 x + (x - 1) \cdot - 6}$

Multiply $- 1$ by $5$ .

$f (\frac{5 + 6 x}{x - 1}) = \frac{5 + 6 x + 5 \cdot x - 5}{5 + 6 x + (x - 1) \cdot - 6}$

Subtract $5$ from $5$ .

$f (\frac{5 + 6 x}{x - 1}) = \frac{6 x + 5 x + 0}{5 + 6 x + (x - 1) \cdot - 6}$

Add $6 x + 5 x$ and $0$ .

$f (\frac{5 + 6 x}{x - 1}) = \frac{6 x + 5 x}{5 + 6 x + (x - 1) \cdot - 6}$

Add $6 x$ and $5 x$ .

$f (\frac{5 + 6 x}{x - 1}) = \frac{11 x}{5 + 6 x + (x - 1) \cdot - 6}$

Simplify the denominator.

Apply the distributive property.

$f (\frac{5 + 6 x}{x - 1}) = \frac{11 x}{5 + 6 x + x \cdot - 6 - 1 \cdot - 6}$

Move $- 6$ to the left of $x$ .

$f (\frac{5 + 6 x}{x - 1}) = \frac{11 x}{5 + 6 x - 6 \cdot x - 1 \cdot - 6}$

Multiply $- 1$ by $- 6$ .

$f (\frac{5 + 6 x}{x - 1}) = \frac{11 x}{5 + 6 x - 6 \cdot x + 6}$

Add $5$ and $6$ .

$f (\frac{5 + 6 x}{x - 1}) = \frac{11 x}{6 x - 6 x + 11}$

Subtract $6 x$ from $6 x$ .

$f (\frac{5 + 6 x}{x - 1}) = \frac{11 x}{0 + 11}$

Add $0$ and $11$ .

$f (\frac{5 + 6 x}{x - 1}) = \frac{11 x}{11}$

Cancel the common factor of $11$ .

Cancel the common factor.

$f (\frac{5 + 6 x}{x - 1}) = \frac{11 x}{11}$

Divide $x$ by $1$ .

$f (\frac{5 + 6 x}{x - 1}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{5 + 6 x}{x - 1}$ is the inverse of $f (x) = \frac{x + 5}{x - 6}$ .

$f^{- 1} (x) = \frac{5 + 6 x}{x - 1}$

Find the Inverse (x+5)/(x-6)

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