Find the Inverse f(x)=(x+12)/(x-6)

$f (x) = \frac{x + 12}{x - 6}$

Step 1

Write $f (x) = \frac{x + 12}{x - 6}$ as an equation.

$y = \frac{x + 12}{x - 6}$

Step 2

Interchange the variables.

$x = \frac{y + 12}{y - 6}$

Step 3

Solve for $y$ .

Multiply the equation by $y - 6$ .

$(y - 6) (x) = (\frac{y + 12}{y - 6}) (y - 6)$

Simplify the left side.

Apply the distributive property.

$y x - 6 x = (\frac{y + 12}{y - 6}) (y - 6)$

Simplify the right side.

Cancel the common factor of $y - 6$ .

Cancel the common factor.

$y x - 6 x = \frac{y + 12}{y - 6} (y - 6)$

Rewrite the expression.

$y x - 6 x = y + 12$

Solve for $y$ .

Subtract $y$ from both sides of the equation.

$y x - 6 x - y = 12$

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

$y x - y = 12 + 6 x$

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

Factor $y$ out of $y x$ .

$y (x) - y = 12 + 6 x$

Factor $y$ out of $- y$ .

$y (x) + y \cdot - 1 = 12 + 6 x$

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

$y (x - 1) = 12 + 6 x$

Divide each term in $y (x - 1) = 12 + 6 x$ by $x - 1$ and simplify.

Divide each term in $y (x - 1) = 12 + 6 x$ by $x - 1$ .

$\frac{y (x - 1)}{x - 1} = \frac{12}{x - 1} + \frac{6 x}{x - 1}$

Simplify the left side.

Cancel the common factor of $x - 1$ .

Cancel the common factor.

$\frac{y (x - 1)}{x - 1} = \frac{12}{x - 1} + \frac{6 x}{x - 1}$

Divide $y$ by $1$ .

$y = \frac{12}{x - 1} + \frac{6 x}{x - 1}$

Simplify the right side.

Combine the numerators over the common denominator.

$y = \frac{12 + 6 x}{x - 1}$

Factor $6$ out of $12 + 6 x$ .

Factor $6$ out of $12$ .

$y = \frac{6 \cdot 2 + 6 x}{x - 1}$

Factor $6$ out of $6 \cdot 2 + 6 x$ .

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

Step 4

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

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

Step 5

Verify if $f^{- 1} (x) = \frac{6 (2 + x)}{x - 1}$ is the inverse of $f (x) = \frac{x + 12}{x - 6}$ .

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 $f^{- 1} (\frac{x + 12}{x - 6})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{x + 12}{x - 6}) = \frac{6 (2 + \frac{x + 12}{x - 6})}{(\frac{x + 12}{x - 6}) - 1}$

Remove parentheses.

$f^{- 1} (\frac{x + 12}{x - 6}) = \frac{6 (2 + \frac{x + 12}{x - 6})}{(\frac{x + 12}{x - 6}) - 1}$

Simplify the numerator.

To write $2$ as a fraction with a common denominator, multiply by $\frac{x - 6}{x - 6}$ .

$f^{- 1} (\frac{x + 12}{x - 6}) = \frac{6 (\frac{2 (x - 6)}{x - 6} + \frac{x + 12}{x - 6})}{\frac{x + 12}{x - 6} - 1}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{x + 12}{x - 6}) = \frac{6 (\frac{2 (x - 6) + x + 12}{x - 6})}{\frac{x + 12}{x - 6} - 1}$

Reorder terms.

$f^{- 1} (\frac{x + 12}{x - 6}) = \frac{6 (\frac{x + 2 (x - 6) + 12}{x - 6})}{\frac{x + 12}{x - 6} - 1}$

Rewrite $\frac{x + 2 (x - 6) + 12}{x - 6}$ in a factored form.

Apply the distributive property.

$f^{- 1} (\frac{x + 12}{x - 6}) = \frac{6 (\frac{x + 2 x + 2 \cdot - 6 + 12}{x - 6})}{\frac{x + 12}{x - 6} - 1}$

Multiply $2$ by $- 6$ .

$f^{- 1} (\frac{x + 12}{x - 6}) = \frac{6 (\frac{x + 2 x - 12 + 12}{x - 6})}{\frac{x + 12}{x - 6} - 1}$

Add $x$ and $2 x$ .

$f^{- 1} (\frac{x + 12}{x - 6}) = \frac{6 (\frac{3 x - 12 + 12}{x - 6})}{\frac{x + 12}{x - 6} - 1}$

Add $- 12$ and $12$ .

$f^{- 1} (\frac{x + 12}{x - 6}) = \frac{6 (\frac{3 x + 0}{x - 6})}{\frac{x + 12}{x - 6} - 1}$

Add $3 x$ and $0$ .

$f^{- 1} (\frac{x + 12}{x - 6}) = \frac{6 (\frac{3 x}{x - 6})}{\frac{x + 12}{x - 6} - 1}$

Simplify the denominator.

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

$f^{- 1} (\frac{x + 12}{x - 6}) = \frac{6 (\frac{3 x}{x - 6})}{\frac{x + 12}{x - 6} - 1 \cdot \frac{x - 6}{x - 6}}$

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

$f^{- 1} (\frac{x + 12}{x - 6}) = \frac{6 (\frac{3 x}{x - 6})}{\frac{x + 12}{x - 6} + \frac{- (x - 6)}{x - 6}}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{x + 12}{x - 6}) = \frac{6 (\frac{3 x}{x - 6})}{\frac{x + 12 - (x - 6)}{x - 6}}$

Rewrite $\frac{x + 12 - (x - 6)}{x - 6}$ in a factored form.

Apply the distributive property.

$f^{- 1} (\frac{x + 12}{x - 6}) = \frac{6 (\frac{3 x}{x - 6})}{\frac{x + 12 - x + 6}{x - 6}}$

Multiply $- 1$ by $- 6$ .

$f^{- 1} (\frac{x + 12}{x - 6}) = \frac{6 (\frac{3 x}{x - 6})}{\frac{x + 12 - x + 6}{x - 6}}$

Subtract $x$ from $x$ .

$f^{- 1} (\frac{x + 12}{x - 6}) = \frac{6 (\frac{3 x}{x - 6})}{\frac{0 + 12 + 6}{x - 6}}$

Add $0$ and $12$ .

$f^{- 1} (\frac{x + 12}{x - 6}) = \frac{6 (\frac{3 x}{x - 6})}{\frac{12 + 6}{x - 6}}$

Add $12$ and $6$ .

$f^{- 1} (\frac{x + 12}{x - 6}) = \frac{6 (\frac{3 x}{x - 6})}{\frac{18}{x - 6}}$

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

$f^{- 1} (\frac{x + 12}{x - 6}) = \frac{\frac{6 (3 x)}{x - 6}}{\frac{18}{x - 6}}$

Multiply $6$ by $3$ .

$f^{- 1} (\frac{x + 12}{x - 6}) = \frac{\frac{18 x}{x - 6}}{\frac{18}{x - 6}}$

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{x + 12}{x - 6}) = \frac{18 x}{x - 6} \cdot \frac{x - 6}{18}$

Cancel the common factor of $18$ .

Factor $18$ out of $18 x$ .

$f^{- 1} (\frac{x + 12}{x - 6}) = \frac{18 (x)}{x - 6} \cdot \frac{x - 6}{18}$

Cancel the common factor.

$f^{- 1} (\frac{x + 12}{x - 6}) = \frac{18 x}{x - 6} \cdot \frac{x - 6}{18}$

Rewrite the expression.

$f^{- 1} (\frac{x + 12}{x - 6}) = \frac{x}{x - 6} \cdot (x - 6)$

Cancel the common factor of $x - 6$ .

Cancel the common factor.

$f^{- 1} (\frac{x + 12}{x - 6}) = \frac{x}{x - 6} \cdot (x - 6)$

Rewrite the expression.

$f^{- 1} (\frac{x + 12}{x - 6}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{6 (2 + x)}{x - 1})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{6 (2 + x)}{x - 1}) = \frac{(\frac{6 (2 + x)}{x - 1}) + 12}{(\frac{6 (2 + x)}{x - 1}) - 6}$

Cancel the common factor of $(\frac{6 (2 + x)}{x - 1}) + 12$ and $(\frac{6 (2 + x)}{x - 1}) - 6$ .

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

$f (\frac{6 (2 + x)}{x - 1}) = \frac{6 (\frac{2 + x}{x - 1}) + 12}{(\frac{6 (2 + x)}{x - 1}) - 6}$

Factor $6$ out of $12$ .

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

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

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

Cancel the common factors.

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

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

Factor $6$ out of $- 6$ .

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

Multiply $\frac{\frac{2 + x}{x - 1} + 2}{\frac{2 + x}{x - 1} - 1}$ by $\frac{x - 1}{x - 1}$ .

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

Combine.

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

Apply the distributive property.

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

Simplify by cancelling.

Cancel the common factor of $x - 1$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x - 1$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the numerator.

Apply the distributive property.

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

Move $2$ to the left of $x$ .

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

Multiply $- 1$ by $2$ .

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

Subtract $2$ from $2$ .

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

Add $x + 2 x$ and $0$ .

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

Add $x$ and $2 x$ .

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

Simplify the denominator.

Apply the distributive property.

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

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

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

Multiply $- 1$ by $- 1$ .

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

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

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

Add $2$ and $1$ .

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

Subtract $x$ from $x$ .

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

Add $0$ and $3$ .

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{6 (2 + x)}{x - 1}$ is the inverse of $f (x) = \frac{x + 12}{x - 6}$ .

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

Find the Inverse f(x)=(x+12)/(x-6)

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