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

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

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

Multiply the equation by $y - 6$ .

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

Simplify the left side.

Apply the distributive property.

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

Simplify the right side.

Cancel the common factor of $y - 6$ .

Cancel the common factor.

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

Rewrite the expression.

$y x - 6 x = y + 8$

Solve for $y$ .

Subtract $y$ from both sides of the equation.

$y x - 6 x - y = 8$

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

$y x - y = 8 + 6 x$

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

Factor $y$ out of $y x$ .

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

Factor $y$ out of $- y$ .

$y (x) + y \cdot - 1 = 8 + 6 x$

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

$y (x - 1) = 8 + 6 x$

Divide each term in $y (x - 1) = 8 + 6 x$ by $x - 1$ and simplify.

Divide each term in $y (x - 1) = 8 + 6 x$ by $x - 1$ .

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

Simplify the left side.

Cancel the common factor of $x - 1$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Combine the numerators over the common denominator.

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

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

Factor $2$ out of $8$ .

$y = \frac{2 (4) + 6 x}{x - 1}$

Factor $2$ out of $6 x$ .

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

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

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

Step 3

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

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

Step 4

Verify if $f^{- 1} (x) = \frac{2 (4 + 3 x)}{x - 1}$ is the inverse of $f (x) = \frac{x + 8}{x - 6}$ .

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

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

Simplify the numerator.

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

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

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

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

Combine the numerators over the common denominator.

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

Rewrite $\frac{4 (x - 6) + 3 (x + 8)}{x - 6}$ in a factored form.

Apply the distributive property.

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

Multiply $4$ by $- 6$ .

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

Apply the distributive property.

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

Multiply $3$ by $8$ .

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

Add $4 x$ and $3 x$ .

$f^{- 1} (\frac{x + 8}{x - 6}) = \frac{2 (\frac{7 x - 24 + 24}{x - 6})}{\frac{x + 8}{x - 6} - 1}$

Add $- 24$ and $24$ .

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

Add $7 x$ and $0$ .

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

Simplify the denominator.

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

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

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

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

Combine the numerators over the common denominator.

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

Rewrite $\frac{x + 8 - (x - 6)}{x - 6}$ in a factored form.

Apply the distributive property.

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

Multiply $- 1$ by $- 6$ .

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

Subtract $x$ from $x$ .

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

Add $0$ and $8$ .

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

Add $8$ and $6$ .

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

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

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

Multiply $2$ by $7$ .

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

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $14$ .

Factor $14$ out of $14 x$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x - 6$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

Set up the composite result function.

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

Evaluate $f (\frac{2 (4 + 3 x)}{x - 1})$ by substituting in the value of $f^{- 1}$ into $f$ .

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

Cancel the common factor of $(\frac{2 (4 + 3 x)}{x - 1}) + 8$ and $(\frac{2 (4 + 3 x)}{x - 1}) - 6$ .

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

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

Factor $2$ out of $8$ .

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

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

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

Cancel the common factors.

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

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

Factor $2$ out of $- 6$ .

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

Multiply $\frac{\frac{4 + 3 x}{x - 1} + 4}{\frac{4 + 3 x}{x - 1} - 3}$ by $\frac{x - 1}{x - 1}$ .

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

Combine.

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

Apply the distributive property.

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

Simplify by cancelling.

Cancel the common factor of $x - 1$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x - 1$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the numerator.

Apply the distributive property.

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

Move $4$ to the left of $x$ .

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

Multiply $- 1$ by $4$ .

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

Subtract $4$ from $4$ .

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

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

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

Add $3 x$ and $4 x$ .

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

Simplify the denominator.

Apply the distributive property.

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

Move $- 3$ to the left of $x$ .

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

Multiply $- 1$ by $- 3$ .

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

Add $4$ and $3$ .

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

Subtract $3 x$ from $3 x$ .

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

Add $0$ and $7$ .

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

Cancel the common factor of $7$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{2 (4 + 3 x)}{x - 1}$ is the inverse of $f (x) = \frac{x + 8}{x - 6}$ .

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

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

Related Questions

Support LCMGCF.com by disabling your adblocker.