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

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

Step 1

Write $f (x) = \frac{x + 6}{x - 5}$ as an equation.

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

Multiply the equation by $y - 5$ .

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

Simplify the left side.

Apply the distributive property.

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

Simplify the right side.

Cancel the common factor of $y - 5$ .

Cancel the common factor.

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

Rewrite the expression.

$y x - 5 x = y + 6$

Solve for $y$ .

Subtract $y$ from both sides of the equation.

$y x - 5 x - y = 6$

Add $5 x$ to both sides of the equation.

$y x - y = 6 + 5 x$

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

Factor $y$ out of $y x$ .

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

Factor $y$ out of $- y$ .

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

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

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

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

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

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

Simplify the left side.

Cancel the common factor of $x - 1$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Combine the numerators over the common denominator.

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

Step 4

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

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

Step 5

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

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

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

Multiply the numerator and denominator of the complex fraction by $x - 5$ .

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

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

Combine.

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

Apply the distributive property.

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

Cancel the common factor of $x - 5$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the numerator.

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

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

Apply the distributive property.

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

Multiply $6$ by $- 5$ .

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

Apply the distributive property.

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

Multiply $5$ by $6$ .

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

Add $6 x$ and $5 x$ .

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

Add $- 30$ and $30$ .

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

Add $11 x$ and $0$ .

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

Simplify the denominator.

Apply the distributive property.

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

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

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

Multiply $- 5$ by $- 1$ .

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

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

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

Subtract $x$ from $x$ .

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

Add $0$ and $6$ .

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

Add $6$ and $5$ .

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

Reduce the expression by cancelling the common factors.

Factor $\frac{11 x}{x - 5}$ out of $\frac{(x - 5) \frac{11 x}{x - 5}}{11}$ .

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

Cancel the common factor of $11$ .

Factor $11$ out of $11 x$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x - 5$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

Set up the composite result function.

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

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

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

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

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

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

Combine.

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

Apply the distributive property.

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

Simplify by cancelling.

Cancel the common factor of $x - 1$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x - 1$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the numerator.

Apply the distributive property.

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

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

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

Multiply $- 1$ by $6$ .

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

Subtract $6$ from $6$ .

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

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

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

Add $5 x$ and $6 x$ .

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

Simplify the denominator.

Apply the distributive property.

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

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

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

Multiply $- 1$ by $- 5$ .

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

Add $6$ and $5$ .

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

Subtract $5 x$ from $5 x$ .

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

Add $0$ and $11$ .

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

Cancel the common factor of $11$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

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

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

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

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