Find the Inverse f^-1(x)=(2x+6)/(x+4)

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

Write $f^{- 1} (x) = \frac{2 x + 6}{x + 4}$ as an equation.

$y = \frac{2 x + 6}{x + 4}$

Interchange the variables.

$x = \frac{2 y + 6}{y + 4}$

Solve for $y$ .

Rewrite the equation as $\frac{2 y + 6}{y + 4} = x$ .

$\frac{2 y + 6}{y + 4} = x$

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

Factor $2$ out of $2 y$ .

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

Factor $2$ out of $6$ .

$\frac{2 y + 2 \cdot 3}{y + 4} = x$

Factor $2$ out of $2 y + 2 \cdot 3$ .

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

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y + 4, 1$

Remove parentheses.

$y + 4, 1$

The LCM of one and any expression is the expression.

$y + 4$

Multiply each term in $\frac{2 (y + 3)}{y + 4} = x$ by $y + 4$ to eliminate the fractions.

Multiply each term in $\frac{2 (y + 3)}{y + 4} = x$ by $y + 4$ .

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

Simplify the left side.

Cancel the common factor of $y + 4$ .

Cancel the common factor.

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

Rewrite the expression.

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

Apply the distributive property.

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

Multiply $2$ by $3$ .

$2 y + 6 = x (y + 4)$

Simplify the right side.

Apply the distributive property.

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

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

$2 y + 6 = x y + 4 x$

Solve the equation.

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

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

Subtract $6$ from both sides of the equation.

$2 y - x y = 4 x - 6$

Factor $y$ out of $2 y - x y$ .

Factor $y$ out of $2 y$ .

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

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

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

Factor $y$ out of $y \cdot 2 + y (- x)$ .

$y (2 - x) = 4 x - 6$

Divide each term in $y (2 - x) = 4 x - 6$ by $2 - x$ and simplify.

Divide each term in $y (2 - x) = 4 x - 6$ by $2 - x$ .

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

Simplify the left side.

Cancel the common factor of $2 - x$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Combine the numerators over the common denominator.

$y = \frac{4 x - 6}{2 - x}$

Factor $2$ out of $4 x - 6$ .

Factor $2$ out of $4 x$ .

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

Factor $2$ out of $- 6$ .

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

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

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

Replace $y$ with $f (x)$ to show the final answer.

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

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

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

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

Set up the composite result function.

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

Evaluate $f (\frac{2 x + 6}{x + 4})$ by substituting in the value of $f^{- 1}$ into $f$ .

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

Simplify with factoring out.

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

Factor $2$ out of $2 x$ .

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

Factor $2$ out of $6$ .

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

Factor $2$ out of $2 x + 2 \cdot 3$ .

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

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

Factor $2$ out of $2 x$ .

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

Factor $2$ out of $6$ .

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

Factor $2$ out of $2 x + 2 \cdot 3$ .

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

Simplify the numerator.

Multiply $2 \frac{2 (x + 3)}{x + 4}$ .

Combine $2$ and $\frac{2 (x + 3)}{x + 4}$ .

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

Multiply $2$ by $2$ .

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

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

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

Combine $- 3$ and $\frac{x + 4}{x + 4}$ .

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

Combine the numerators over the common denominator.

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

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

Apply the distributive property.

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

Multiply $4$ by $3$ .

$f (\frac{2 x + 6}{x + 4}) = \frac{2 (\frac{4 x + 12 - 3 (x + 4)}{x + 4})}{2 - \frac{2 (x + 3)}{x + 4}}$

Apply the distributive property.

$f (\frac{2 x + 6}{x + 4}) = \frac{2 (\frac{4 x + 12 - 3 x - 3 \cdot 4}{x + 4})}{2 - \frac{2 (x + 3)}{x + 4}}$

Multiply $- 3$ by $4$ .

$f (\frac{2 x + 6}{x + 4}) = \frac{2 (\frac{4 x + 12 - 3 x - 12}{x + 4})}{2 - \frac{2 (x + 3)}{x + 4}}$

Subtract $3 x$ from $4 x$ .

$f (\frac{2 x + 6}{x + 4}) = \frac{2 (\frac{x + 12 - 12}{x + 4})}{2 - \frac{2 (x + 3)}{x + 4}}$

Subtract $12$ from $12$ .

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

Add $x$ and $0$ .

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

Simplify the denominator.

To write $2$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

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

Combine the numerators over the common denominator.

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

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

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

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

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

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

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

Apply the distributive property.

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

Multiply $- 1$ by $3$ .

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

Subtract $x$ from $x$ .

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

Add $0$ and $4$ .

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

Subtract $3$ from $4$ .

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

Multiply $2$ by $1$ .

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

Combine $2$ and $\frac{x}{x + 4}$ .

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

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $2$ .

Factor $2$ out of $2 x$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x + 4$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

Set up the composite result function.

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

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

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

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

Factor $2$ out of $6$ .

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

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

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

Cancel the common factors.

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

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

Factor $2$ out of $4$ .

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the numerator and denominator of the complex fraction by $2 - x$ .

Multiply $\frac{\frac{2 (2 x - 3)}{2 - x} + 3}{\frac{2 x - 3}{2 - x} + 2}$ by $\frac{2 - x}{2 - x}$ .

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

Combine.

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

Apply the distributive property.

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

Simplify by cancelling.

Cancel the common factor of $2 - x$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $2 - x$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $2$ by $2$ .

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

Multiply $2$ by $- 3$ .

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

Apply the distributive property.

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

Multiply $2$ by $3$ .

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

Multiply $3$ by $- 1$ .

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

Subtract $3 x$ from $4 x$ .

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

Add $- 6$ and $6$ .

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

Add $x$ and $0$ .

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

Simplify the denominator.

Apply the distributive property.

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

Multiply $2$ by $2$ .

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

Multiply $2$ by $- 1$ .

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

Subtract $2 x$ from $2 x$ .

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

Subtract $3$ from $0$ .

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

Add $- 3$ and $4$ .

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

Divide $x$ by $1$ .

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

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

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

Find the Inverse f^-1(x)=(2x+6)/(x+4)

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