Find the Inverse (6x+8)/(x-1)

$\frac{6 x + 8}{x - 1}$

Step 1

Interchange the variables.

$x = \frac{6 y + 8}{y - 1}$

Step 2

Solve for $y$ .

Multiply the equation by $y - 1$ .

$(y - 1) (x) = (\frac{6 y + 8}{y - 1}) (y - 1)$

Simplify the left side.

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

Apply the distributive property.

$y x - 1 x = (\frac{6 y + 8}{y - 1}) (y - 1)$

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

$y x - x = (\frac{6 y + 8}{y - 1}) (y - 1)$

Simplify the right side.

Simplify $(\frac{6 y + 8}{y - 1}) (y - 1)$ .

Factor $2$ out of $6 y + 8$ .

Factor $2$ out of $6 y$ .

$y x - x = \frac{2 (3 y) + 8}{y - 1} (y - 1)$

Factor $2$ out of $8$ .

$y x - x = \frac{2 (3 y) + 2 (4)}{y - 1} (y - 1)$

Factor $2$ out of $2 (3 y) + 2 (4)$ .

$y x - x = \frac{2 (3 y + 4)}{y - 1} (y - 1)$

Cancel the common factor of $y - 1$ .

Cancel the common factor.

$y x - x = \frac{2 (3 y + 4)}{y - 1} (y - 1)$

Rewrite the expression.

$y x - x = 2 (3 y + 4)$

Apply the distributive property.

$y x - x = 2 (3 y) + 2 \cdot 4$

Multiply.

Multiply $3$ by $2$ .

$y x - x = 6 y + 2 \cdot 4$

Multiply $2$ by $4$ .

$y x - x = 6 y + 8$

Solve for $y$ .

Subtract $6 y$ from both sides of the equation.

$y x - x - 6 y = 8$

Add $x$ to both sides of the equation.

$y x - 6 y = 8 + x$

Factor $y$ out of $y x - 6 y$ .

Factor $y$ out of $y x$ .

$y (x) - 6 y = 8 + x$

Factor $y$ out of $- 6 y$ .

$y (x) + y \cdot - 6 = 8 + x$

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

$y (x - 6) = 8 + x$

Divide each term in $y (x - 6) = 8 + x$ by $x - 6$ and simplify.

Divide each term in $y (x - 6) = 8 + x$ by $x - 6$ .

$\frac{y (x - 6)}{x - 6} = \frac{8}{x - 6} + \frac{x}{x - 6}$

Simplify the left side.

Cancel the common factor of $x - 6$ .

Cancel the common factor.

$\frac{y (x - 6)}{x - 6} = \frac{8}{x - 6} + \frac{x}{x - 6}$

Divide $y$ by $1$ .

$y = \frac{8}{x - 6} + \frac{x}{x - 6}$

Simplify the right side.

Combine the numerators over the common denominator.

$y = \frac{8 + x}{x - 6}$

Step 3

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

$f^{- 1} (x) = \frac{8 + x}{x - 6}$

Step 4

Verify if $f^{- 1} (x) = \frac{8 + x}{x - 6}$ is the inverse of $f (x) = \frac{6 x + 8}{x - 1}$ .

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 $f^{- 1} (\frac{6 x + 8}{x - 1})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{8 + \frac{6 x + 8}{x - 1}}{(\frac{6 x + 8}{x - 1}) - 6}$

Remove parentheses.

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{8 + \frac{6 x + 8}{x - 1}}{(\frac{6 x + 8}{x - 1}) - 6}$

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

Multiply $\frac{8 + \frac{6 x + 8}{x - 1}}{\frac{6 x + 8}{x - 1} - 6}$ by $\frac{x - 1}{x - 1}$ .

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{x - 1}{x - 1} \cdot \frac{8 + \frac{6 x + 8}{x - 1}}{\frac{6 x + 8}{x - 1} - 6}$

Combine.

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{(x - 1) (8 + \frac{6 x + 8}{x - 1})}{(x - 1) (\frac{6 x + 8}{x - 1} - 6)}$

Apply the distributive property.

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{(x - 1) \cdot 8 + (x - 1) (\frac{6 x + 8}{x - 1})}{(x - 1) (\frac{6 x + 8}{x - 1}) + (x - 1) \cdot - 6}$

Simplify by cancelling.

Factor $2$ out of $6 x + 8$ .

Factor $2$ out of $6 x$ .

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{(x - 1) \cdot 8 + (x - 1) (\frac{2 (3 x) + 8}{x - 1})}{(x - 1) (\frac{6 x + 8}{x - 1}) + (x - 1) \cdot - 6}$

Factor $2$ out of $8$ .

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{(x - 1) \cdot 8 + (x - 1) (\frac{2 (3 x) + 2 (4)}{x - 1})}{(x - 1) (\frac{6 x + 8}{x - 1}) + (x - 1) \cdot - 6}$

Factor $2$ out of $2 (3 x) + 2 (4)$ .

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{(x - 1) \cdot 8 + (x - 1) (\frac{2 (3 x + 4)}{x - 1})}{(x - 1) (\frac{6 x + 8}{x - 1}) + (x - 1) \cdot - 6}$

Factor $2$ out of $6 x + 8$ .

Factor $2$ out of $6 x$ .

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{(x - 1) \cdot 8 + (x - 1) (\frac{2 (3 x + 4)}{x - 1})}{(x - 1) (\frac{2 (3 x) + 8}{x - 1}) + (x - 1) \cdot - 6}$

Factor $2$ out of $8$ .

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{(x - 1) \cdot 8 + (x - 1) (\frac{2 (3 x + 4)}{x - 1})}{(x - 1) (\frac{2 (3 x) + 2 (4)}{x - 1}) + (x - 1) \cdot - 6}$

Factor $2$ out of $2 (3 x) + 2 (4)$ .

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{(x - 1) \cdot 8 + (x - 1) (\frac{2 (3 x + 4)}{x - 1})}{(x - 1) (\frac{2 (3 x + 4)}{x - 1}) + (x - 1) \cdot - 6}$

Simplify the numerator.

Factor $x - 1$ out of $(x - 1) \cdot 8 + (x - 1) \frac{2 (3 x + 4)}{x - 1}$ .

Factor $x - 1$ out of $(x - 1) \cdot 8$ .

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{(x - 1) (8) + (x - 1) (\frac{2 (3 x + 4)}{x - 1})}{(x - 1) (\frac{2 (3 x + 4)}{x - 1}) + (x - 1) \cdot - 6}$

Factor $x - 1$ out of $(x - 1) (8) + (x - 1) (\frac{2 (3 x + 4)}{x - 1})$ .

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{(x - 1) (8 + \frac{2 (3 x + 4)}{x - 1})}{(x - 1) (\frac{2 (3 x + 4)}{x - 1}) + (x - 1) \cdot - 6}$

To write $8$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{(x - 1) (\frac{8 (x - 1)}{x - 1} + \frac{2 (3 x + 4)}{x - 1})}{(x - 1) (\frac{2 (3 x + 4)}{x - 1}) + (x - 1) \cdot - 6}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{(x - 1) (\frac{8 (x - 1) + 2 (3 x + 4)}{x - 1})}{(x - 1) (\frac{2 (3 x + 4)}{x - 1}) + (x - 1) \cdot - 6}$

Rewrite $\frac{8 (x - 1) + 2 (3 x + 4)}{x - 1}$ in a factored form.

Factor $2$ out of $8 (x - 1) + 2 (3 x + 4)$ .

Factor $2$ out of $8 (x - 1)$ .

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{(x - 1) (\frac{2 (4 (x - 1)) + 2 (3 x + 4)}{x - 1})}{(x - 1) (\frac{2 (3 x + 4)}{x - 1}) + (x - 1) \cdot - 6}$

Factor $2$ out of $2 (4 (x - 1)) + 2 (3 x + 4)$ .

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{(x - 1) (\frac{2 (4 (x - 1) + 3 x + 4)}{x - 1})}{(x - 1) (\frac{2 (3 x + 4)}{x - 1}) + (x - 1) \cdot - 6}$

Apply the distributive property.

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{(x - 1) (\frac{2 (4 x + 4 \cdot - 1 + 3 x + 4)}{x - 1})}{(x - 1) (\frac{2 (3 x + 4)}{x - 1}) + (x - 1) \cdot - 6}$

Multiply $4$ by $- 1$ .

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{(x - 1) (\frac{2 (4 x - 4 + 3 x + 4)}{x - 1})}{(x - 1) (\frac{2 (3 x + 4)}{x - 1}) + (x - 1) \cdot - 6}$

Add $4 x$ and $3 x$ .

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{(x - 1) (\frac{2 (7 x - 4 + 4)}{x - 1})}{(x - 1) (\frac{2 (3 x + 4)}{x - 1}) + (x - 1) \cdot - 6}$

Add $- 4$ and $4$ .

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{(x - 1) (\frac{2 (7 x + 0)}{x - 1})}{(x - 1) (\frac{2 (3 x + 4)}{x - 1}) + (x - 1) \cdot - 6}$

Add $7 x$ and $0$ .

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{(x - 1) (\frac{2 \cdot (7 x)}{x - 1})}{(x - 1) (\frac{2 (3 x + 4)}{x - 1}) + (x - 1) \cdot - 6}$

Multiply $2$ by $7$ .

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{(x - 1) (\frac{14 x}{x - 1})}{(x - 1) (\frac{2 (3 x + 4)}{x - 1}) + (x - 1) \cdot - 6}$

Simplify the denominator.

Factor $x - 1$ out of $(x - 1) \frac{2 (3 x + 4)}{x - 1} + (x - 1) \cdot - 6$ .

Factor $x - 1$ out of $(x - 1) \frac{2 (3 x + 4)}{x - 1}$ .

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{(x - 1) (\frac{14 x}{x - 1})}{(x - 1) (\frac{2 (3 x + 4)}{x - 1}) + (x - 1) \cdot - 6}$

Factor $x - 1$ out of $(x - 1) \cdot - 6$ .

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{(x - 1) (\frac{14 x}{x - 1})}{(x - 1) (\frac{2 (3 x + 4)}{x - 1}) + (x - 1) (- 6)}$

Factor $x - 1$ out of $(x - 1) (\frac{2 (3 x + 4)}{x - 1}) + (x - 1) (- 6)$ .

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{(x - 1) (\frac{14 x}{x - 1})}{(x - 1) (\frac{2 (3 x + 4)}{x - 1} - 6)}$

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{(x - 1) (\frac{14 x}{x - 1})}{(x - 1) (\frac{2 (3 x + 4)}{x - 1} - 6 \cdot \frac{x - 1}{x - 1})}$

Combine $- 6$ and $\frac{x - 1}{x - 1}$ .

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{(x - 1) (\frac{14 x}{x - 1})}{(x - 1) (\frac{2 (3 x + 4)}{x - 1} + \frac{- 6 (x - 1)}{x - 1})}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{(x - 1) (\frac{14 x}{x - 1})}{(x - 1) (\frac{2 (3 x + 4) - 6 (x - 1)}{x - 1})}$

Rewrite $\frac{2 (3 x + 4) - 6 (x - 1)}{x - 1}$ in a factored form.

Factor $2$ out of $2 (3 x + 4) - 6 (x - 1)$ .

Factor $2$ out of $- 6 (x - 1)$ .

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{(x - 1) (\frac{14 x}{x - 1})}{(x - 1) (\frac{2 (3 x + 4) + 2 (- 3 (x - 1))}{x - 1})}$

Factor $2$ out of $2 (3 x + 4) + 2 (- 3 (x - 1))$ .

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{(x - 1) (\frac{14 x}{x - 1})}{(x - 1) (\frac{2 (3 x + 4 - 3 (x - 1))}{x - 1})}$

Apply the distributive property.

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{(x - 1) (\frac{14 x}{x - 1})}{(x - 1) (\frac{2 (3 x + 4 - 3 x - 3 \cdot - 1)}{x - 1})}$

Multiply $- 3$ by $- 1$ .

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{(x - 1) (\frac{14 x}{x - 1})}{(x - 1) (\frac{2 (3 x + 4 - 3 x + 3)}{x - 1})}$

Subtract $3 x$ from $3 x$ .

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{(x - 1) (\frac{14 x}{x - 1})}{(x - 1) (\frac{2 (0 + 4 + 3)}{x - 1})}$

Add $0$ and $4$ .

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{(x - 1) (\frac{14 x}{x - 1})}{(x - 1) (\frac{2 (4 + 3)}{x - 1})}$

Add $4$ and $3$ .

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{(x - 1) (\frac{14 x}{x - 1})}{(x - 1) (\frac{2 \cdot 7}{x - 1})}$

Multiply $2$ by $7$ .

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{(x - 1) (\frac{14 x}{x - 1})}{(x - 1) (\frac{14}{x - 1})}$

Cancel the common factor of $x - 1$ .

Cancel the common factor.

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{(x - 1) (\frac{14 x}{x - 1})}{(x - 1) (\frac{14}{x - 1})}$

Rewrite the expression.

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{\frac{14 x}{x - 1}}{\frac{14}{x - 1}}$

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{14 x}{x - 1} \cdot \frac{x - 1}{14}$

Cancel the common factor of $14$ .

Factor $14$ out of $14 x$ .

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{14 (x)}{x - 1} \cdot \frac{x - 1}{14}$

Cancel the common factor.

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{14 x}{x - 1} \cdot \frac{x - 1}{14}$

Rewrite the expression.

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{x}{x - 1} \cdot (x - 1)$

Cancel the common factor of $x - 1$ .

Cancel the common factor.

$f^{- 1} (\frac{6 x + 8}{x - 1}) = \frac{x}{x - 1} \cdot (x - 1)$

Rewrite the expression.

$f^{- 1} (\frac{6 x + 8}{x - 1}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{8 + x}{x - 6})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{8 + x}{x - 6}) = \frac{6 (\frac{8 + x}{x - 6}) + 8}{(\frac{8 + x}{x - 6}) - 1}$

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

Multiply $\frac{6 \frac{8 + x}{x - 6} + 8}{\frac{8 + x}{x - 6} - 1}$ by $\frac{x - 6}{x - 6}$ .

$f (\frac{8 + x}{x - 6}) = \frac{x - 6}{x - 6} \cdot \frac{6 (\frac{8 + x}{x - 6}) + 8}{\frac{8 + x}{x - 6} - 1}$

Combine.

$f (\frac{8 + x}{x - 6}) = \frac{(x - 6) (6 (\frac{8 + x}{x - 6}) + 8)}{(x - 6) (\frac{8 + x}{x - 6} - 1)}$

Apply the distributive property.

$f (\frac{8 + x}{x - 6}) = \frac{(x - 6) (6 (\frac{8 + x}{x - 6})) + (x - 6) \cdot 8}{(x - 6) (\frac{8 + x}{x - 6}) + (x - 6) \cdot - 1}$

Cancel the common factor of $x - 6$ .

Cancel the common factor.

$f (\frac{8 + x}{x - 6}) = \frac{(x - 6) (6 (\frac{8 + x}{x - 6})) + (x - 6) \cdot 8}{(x - 6) (\frac{8 + x}{x - 6}) + (x - 6) \cdot - 1}$

Rewrite the expression.

$f (\frac{8 + x}{x - 6}) = \frac{(x - 6) (6 (\frac{8 + x}{x - 6})) + (x - 6) \cdot 8}{8 + x + (x - 6) \cdot - 1}$

Simplify the numerator.

Factor $(x - 6) \cdot 2$ out of $(x - 6) \cdot 6 \frac{8 + x}{x - 6} + (x - 6) \cdot 8$ .

Factor $(x - 6) \cdot 2$ out of $(x - 6) \cdot 6 \frac{8 + x}{x - 6}$ .

$f (\frac{8 + x}{x - 6}) = \frac{(x - 6) \cdot (2 (3 (\frac{8 + x}{x - 6}))) + (x - 6) \cdot 8}{8 + x + (x - 6) \cdot - 1}$

Factor $(x - 6) \cdot 2$ out of $(x - 6) \cdot 8$ .

$f (\frac{8 + x}{x - 6}) = \frac{(x - 6) \cdot (2 (3 (\frac{8 + x}{x - 6}))) + (x - 6) \cdot (2 (4))}{8 + x + (x - 6) \cdot - 1}$

Factor $(x - 6) \cdot 2$ out of $(x - 6) \cdot 2 (3 \frac{8 + x}{x - 6}) + (x - 6) \cdot 2 (4)$ .

$f (\frac{8 + x}{x - 6}) = \frac{(x - 6) \cdot (2 (3 (\frac{8 + x}{x - 6}) + 4))}{8 + x + (x - 6) \cdot - 1}$

Combine $3$ and $\frac{8 + x}{x - 6}$ .

$f (\frac{8 + x}{x - 6}) = \frac{(x - 6) \cdot (2 (\frac{3 (8 + x)}{x - 6} + 4))}{8 + x + (x - 6) \cdot - 1}$

To write $4$ as a fraction with a common denominator, multiply by $\frac{x - 6}{x - 6}$ .

$f (\frac{8 + x}{x - 6}) = \frac{(x - 6) \cdot (2 (\frac{3 (8 + x)}{x - 6} + \frac{4 (x - 6)}{x - 6}))}{8 + x + (x - 6) \cdot - 1}$

Combine the numerators over the common denominator.

$f (\frac{8 + x}{x - 6}) = \frac{(x - 6) \cdot (2 (\frac{3 (8 + x) + 4 (x - 6)}{x - 6}))}{8 + x + (x - 6) \cdot - 1}$

Simplify the numerator.

Apply the distributive property.

$f (\frac{8 + x}{x - 6}) = \frac{(x - 6) \cdot (2 (\frac{3 \cdot 8 + 3 x + 4 (x - 6)}{x - 6}))}{8 + x + (x - 6) \cdot - 1}$

Multiply $3$ by $8$ .

$f (\frac{8 + x}{x - 6}) = \frac{(x - 6) \cdot (2 (\frac{24 + 3 x + 4 (x - 6)}{x - 6}))}{8 + x + (x - 6) \cdot - 1}$

Apply the distributive property.

$f (\frac{8 + x}{x - 6}) = \frac{(x - 6) \cdot (2 (\frac{24 + 3 x + 4 x + 4 \cdot - 6}{x - 6}))}{8 + x + (x - 6) \cdot - 1}$

Multiply $4$ by $- 6$ .

$f (\frac{8 + x}{x - 6}) = \frac{(x - 6) \cdot (2 (\frac{24 + 3 x + 4 x - 24}{x - 6}))}{8 + x + (x - 6) \cdot - 1}$

Subtract $24$ from $24$ .

$f (\frac{8 + x}{x - 6}) = \frac{(x - 6) \cdot (2 (\frac{3 x + 4 x + 0}{x - 6}))}{8 + x + (x - 6) \cdot - 1}$

Add $3 x + 4 x$ and $0$ .

$f (\frac{8 + x}{x - 6}) = \frac{(x - 6) \cdot (2 (\frac{3 x + 4 x}{x - 6}))}{8 + x + (x - 6) \cdot - 1}$

Add $3 x$ and $4 x$ .

$f (\frac{8 + x}{x - 6}) = \frac{(x - 6) \cdot (2 (\frac{7 x}{x - 6}))}{8 + x + (x - 6) \cdot - 1}$

Combine exponents.

Combine $\frac{7 x}{x - 6}$ and $2$ .

$f (\frac{8 + x}{x - 6}) = \frac{(x - 6) \cdot \frac{7 x \cdot 2}{x - 6}}{8 + x + (x - 6) \cdot - 1}$

Multiply $2$ by $7$ .

$f (\frac{8 + x}{x - 6}) = \frac{(x - 6) \cdot \frac{14 x}{x - 6}}{8 + x + (x - 6) \cdot - 1}$

Simplify the denominator.

Apply the distributive property.

$f (\frac{8 + x}{x - 6}) = \frac{(x - 6) \cdot \frac{14 x}{x - 6}}{8 + x + x \cdot - 1 - 6 \cdot - 1}$

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

$f (\frac{8 + x}{x - 6}) = \frac{(x - 6) \cdot \frac{14 x}{x - 6}}{8 + x - 1 \cdot x - 6 \cdot - 1}$

Multiply $- 6$ by $- 1$ .

$f (\frac{8 + x}{x - 6}) = \frac{(x - 6) \cdot \frac{14 x}{x - 6}}{8 + x - 1 \cdot x + 6}$

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

$f (\frac{8 + x}{x - 6}) = \frac{(x - 6) \cdot \frac{14 x}{x - 6}}{8 + x - x + 6}$

Add $8$ and $6$ .

$f (\frac{8 + x}{x - 6}) = \frac{(x - 6) \cdot \frac{14 x}{x - 6}}{x - x + 14}$

Subtract $x$ from $x$ .

$f (\frac{8 + x}{x - 6}) = \frac{(x - 6) \cdot \frac{14 x}{x - 6}}{0 + 14}$

Add $0$ and $14$ .

$f (\frac{8 + x}{x - 6}) = \frac{(x - 6) \cdot \frac{14 x}{x - 6}}{14}$

Reduce the expression by cancelling the common factors.

Factor $\frac{14 x}{x - 6}$ out of $\frac{(x - 6) \cdot \frac{14 x}{x - 6}}{14}$ .

$f (\frac{8 + x}{x - 6}) = \frac{14 x}{x - 6} \cdot \frac{x - 6}{14}$

Cancel the common factor of $14$ .

Factor $14$ out of $14 x$ .

$f (\frac{8 + x}{x - 6}) = \frac{14 (x)}{x - 6} \cdot \frac{x - 6}{14}$

Cancel the common factor.

$f (\frac{8 + x}{x - 6}) = \frac{14 x}{x - 6} \cdot \frac{x - 6}{14}$

Rewrite the expression.

$f (\frac{8 + x}{x - 6}) = \frac{x}{x - 6} \cdot (x - 6)$

Cancel the common factor of $x - 6$ .

Cancel the common factor.

$f (\frac{8 + x}{x - 6}) = \frac{x}{x - 6} \cdot (x - 6)$

Rewrite the expression.

$f (\frac{8 + x}{x - 6}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{8 + x}{x - 6}$ is the inverse of $f (x) = \frac{6 x + 8}{x - 1}$ .

$f^{- 1} (x) = \frac{8 + x}{x - 6}$

Find the Inverse (6x+8)/(x-1)

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