Find the Inverse (6x-4)/(5-x)

$\frac{6 x - 4}{5 - x}$

Step 1

Interchange the variables.

$x = \frac{6 y - 4}{5 - y}$

Step 2

Solve for $y$ .

Rewrite the equation as $\frac{6 y - 4}{5 - y} = x$ .

$\frac{6 y - 4}{5 - y} = x$

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

Factor $2$ out of $6 y$ .

$\frac{2 (3 y) - 4}{5 - y} = x$

Factor $2$ out of $- 4$ .

$\frac{2 (3 y) + 2 (- 2)}{5 - y} = x$

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

$\frac{2 (3 y - 2)}{5 - y} = x$

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.

$5 - y, 1$

Remove parentheses.

$5 - y, 1$

The LCM of one and any expression is the expression.

$5 - y$

Multiply each term in $\frac{2 (3 y - 2)}{5 - y} = x$ by $5 - y$ to eliminate the fractions.

Multiply each term in $\frac{2 (3 y - 2)}{5 - y} = x$ by $5 - y$ .

$\frac{2 (3 y - 2)}{5 - y} (5 - y) = x (5 - y)$

Simplify the left side.

Cancel the common factor of $5 - y$ .

Cancel the common factor.

$\frac{2 (3 y - 2)}{5 - y} (5 - y) = x (5 - y)$

Rewrite the expression.

$2 (3 y - 2) = x (5 - y)$

Apply the distributive property.

$2 (3 y) + 2 \cdot - 2 = x (5 - y)$

Multiply.

Multiply $3$ by $2$ .

$6 y + 2 \cdot - 2 = x (5 - y)$

Multiply $2$ by $- 2$ .

$6 y - 4 = x (5 - y)$

Simplify the right side.

Apply the distributive property.

$6 y - 4 = x \cdot 5 + x (- y)$

Reorder.

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

$6 y - 4 = 5 \cdot x + x (- y)$

Rewrite using the commutative property of multiplication.

$6 y - 4 = 5 x - x y$

Solve the equation.

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

$6 y - 4 + x y = 5 x$

Add $4$ to both sides of the equation.

$6 y + x y = 5 x + 4$

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

Factor $y$ out of $6 y$ .

$y \cdot 6 + x y = 5 x + 4$

Factor $y$ out of $x y$ .

$y \cdot 6 + y x = 5 x + 4$

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

$y (6 + x) = 5 x + 4$

Divide each term in $y (6 + x) = 5 x + 4$ by $6 + x$ and simplify.

Divide each term in $y (6 + x) = 5 x + 4$ by $6 + x$ .

$\frac{y (6 + x)}{6 + x} = \frac{5 x}{6 + x} + \frac{4}{6 + x}$

Simplify the left side.

Cancel the common factor of $6 + x$ .

Cancel the common factor.

$\frac{y (6 + x)}{6 + x} = \frac{5 x}{6 + x} + \frac{4}{6 + x}$

Divide $y$ by $1$ .

$y = \frac{5 x}{6 + x} + \frac{4}{6 + x}$

Simplify the right side.

Combine the numerators over the common denominator.

$y = \frac{5 x + 4}{6 + x}$

Step 3

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

$f^{- 1} (x) = \frac{5 x + 4}{6 + x}$

Step 4

Verify if $f^{- 1} (x) = \frac{5 x + 4}{6 + x}$ is the inverse of $f (x) = \frac{6 x - 4}{5 - x}$ .

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 $f^{- 1} (\frac{6 x - 4}{5 - x})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{5 (\frac{6 x - 4}{5 - x}) + 4}{6 + \frac{6 x - 4}{5 - x}}$

Remove parentheses.

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{5 (\frac{6 x - 4}{5 - x}) + 4}{6 + \frac{6 x - 4}{5 - x}}$

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

Multiply $\frac{5 \frac{6 x - 4}{5 - x} + 4}{6 + \frac{6 x - 4}{5 - x}}$ by $\frac{5 - x}{5 - x}$ .

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{5 - x}{5 - x} \cdot \frac{5 (\frac{6 x - 4}{5 - x}) + 4}{6 + \frac{6 x - 4}{5 - x}}$

Combine.

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (5 (\frac{6 x - 4}{5 - x}) + 4)}{(5 - x) (6 + \frac{6 x - 4}{5 - x})}$

Apply the distributive property.

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (5 (\frac{6 x - 4}{5 - x})) + (5 - x) \cdot 4}{(5 - x) \cdot 6 + (5 - x) (\frac{6 x - 4}{5 - x})}$

Simplify with factoring out.

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

Factor $2$ out of $6 x$ .

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (5 (\frac{6 x - 4}{5 - x})) + (5 - x) \cdot 4}{(5 - x) \cdot 6 + (5 - x) (\frac{2 (3 x) - 4}{5 - x})}$

Factor $2$ out of $- 4$ .

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (5 (\frac{6 x - 4}{5 - x})) + (5 - x) \cdot 4}{(5 - x) \cdot 6 + (5 - x) (\frac{2 (3 x) + 2 (- 2)}{5 - x})}$

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

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (5 (\frac{6 x - 4}{5 - x})) + (5 - x) \cdot 4}{(5 - x) \cdot 6 + (5 - x) (\frac{2 (3 x - 2)}{5 - x})}$

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

Factor $2$ out of $6 x$ .

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) \cdot (5 (\frac{2 (3 x) - 4}{5 - x})) + (5 - x) \cdot 4}{(5 - x) \cdot 6 + (5 - x) (\frac{2 (3 x - 2)}{5 - x})}$

Factor $2$ out of $- 4$ .

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) \cdot (5 (\frac{2 (3 x) + 2 (- 2)}{5 - x})) + (5 - x) \cdot 4}{(5 - x) \cdot 6 + (5 - x) (\frac{2 (3 x - 2)}{5 - x})}$

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

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) \cdot (5 (\frac{2 (3 x - 2)}{5 - x})) + (5 - x) \cdot 4}{(5 - x) \cdot 6 + (5 - x) (\frac{2 (3 x - 2)}{5 - x})}$

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (5 (\frac{2 (3 x - 2)}{5 - x})) + (5 - x) \cdot 4}{(5 - x) \cdot 6 + (5 - x) (\frac{2 (3 x - 2)}{5 - x})}$

Simplify the numerator.

Factor $5 - x$ out of $(5 - x) \cdot 5 \frac{2 (3 x - 2)}{5 - x} + (5 - x) \cdot 4$ .

Factor $5 - x$ out of $(5 - x) \cdot 4$ .

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (5 (\frac{2 (3 x - 2)}{5 - x})) + (5 - x) (4)}{(5 - x) \cdot 6 + (5 - x) (\frac{2 (3 x - 2)}{5 - x})}$

Factor $5 - x$ out of $(5 - x) (5 \frac{2 (3 x - 2)}{5 - x}) + (5 - x) (4)$ .

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (5 (\frac{2 (3 x - 2)}{5 - x}) + 4)}{(5 - x) \cdot 6 + (5 - x) (\frac{2 (3 x - 2)}{5 - x})}$

Multiply $5 \frac{2 (3 x - 2)}{5 - x}$ .

Combine $5$ and $\frac{2 (3 x - 2)}{5 - x}$ .

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (\frac{5 (2 (3 x - 2))}{5 - x} + 4)}{(5 - x) \cdot 6 + (5 - x) (\frac{2 (3 x - 2)}{5 - x})}$

Multiply $2$ by $5$ .

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (\frac{10 (3 x - 2)}{5 - x} + 4)}{(5 - x) \cdot 6 + (5 - x) (\frac{2 (3 x - 2)}{5 - x})}$

To write $4$ as a fraction with a common denominator, multiply by $\frac{5 - x}{5 - x}$ .

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (\frac{10 (3 x - 2)}{5 - x} + \frac{4 (5 - x)}{5 - x})}{(5 - x) \cdot 6 + (5 - x) (\frac{2 (3 x - 2)}{5 - x})}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (\frac{10 (3 x - 2) + 4 (5 - x)}{5 - x})}{(5 - x) \cdot 6 + (5 - x) (\frac{2 (3 x - 2)}{5 - x})}$

Reorder terms.

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (\frac{10 (3 x - 2) + 4 (5 - x)}{- x + 5})}{(5 - x) \cdot 6 + (5 - x) (\frac{2 (3 x - 2)}{5 - x})}$

Rewrite $\frac{10 (3 x - 2) + 4 (5 - x)}{- x + 5}$ in a factored form.

Factor $2$ out of $10 (3 x - 2) + 4 (5 - x)$ .

Factor $2$ out of $10 (3 x - 2)$ .

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (\frac{2 (5 (3 x - 2)) + 4 (5 - x)}{- x + 5})}{(5 - x) \cdot 6 + (5 - x) (\frac{2 (3 x - 2)}{5 - x})}$

Factor $2$ out of $4 (5 - x)$ .

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (\frac{2 (5 (3 x - 2)) + 2 (2 (5 - x))}{- x + 5})}{(5 - x) \cdot 6 + (5 - x) (\frac{2 (3 x - 2)}{5 - x})}$

Factor $2$ out of $2 (5 (3 x - 2)) + 2 (2 (5 - x))$ .

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (\frac{2 (5 (3 x - 2) + 2 (5 - x))}{- x + 5})}{(5 - x) \cdot 6 + (5 - x) (\frac{2 (3 x - 2)}{5 - x})}$

Apply the distributive property.

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (\frac{2 (5 (3 x) + 5 \cdot - 2 + 2 (5 - x))}{- x + 5})}{(5 - x) \cdot 6 + (5 - x) (\frac{2 (3 x - 2)}{5 - x})}$

Multiply $3$ by $5$ .

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (\frac{2 (15 x + 5 \cdot - 2 + 2 (5 - x))}{- x + 5})}{(5 - x) \cdot 6 + (5 - x) (\frac{2 (3 x - 2)}{5 - x})}$

Multiply $5$ by $- 2$ .

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (\frac{2 (15 x - 10 + 2 (5 - x))}{- x + 5})}{(5 - x) \cdot 6 + (5 - x) (\frac{2 (3 x - 2)}{5 - x})}$

Apply the distributive property.

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (\frac{2 (15 x - 10 + 2 \cdot 5 + 2 (- x))}{- x + 5})}{(5 - x) \cdot 6 + (5 - x) (\frac{2 (3 x - 2)}{5 - x})}$

Multiply $2$ by $5$ .

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (\frac{2 (15 x - 10 + 10 + 2 (- x))}{- x + 5})}{(5 - x) \cdot 6 + (5 - x) (\frac{2 (3 x - 2)}{5 - x})}$

Multiply $- 1$ by $2$ .

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (\frac{2 (15 x - 10 + 10 - 2 x)}{- x + 5})}{(5 - x) \cdot 6 + (5 - x) (\frac{2 (3 x - 2)}{5 - x})}$

Subtract $2 x$ from $15 x$ .

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (\frac{2 (13 x - 10 + 10)}{- x + 5})}{(5 - x) \cdot 6 + (5 - x) (\frac{2 (3 x - 2)}{5 - x})}$

Add $- 10$ and $10$ .

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (\frac{2 (13 x + 0)}{- x + 5})}{(5 - x) \cdot 6 + (5 - x) (\frac{2 (3 x - 2)}{5 - x})}$

Add $13 x$ and $0$ .

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (\frac{2 \cdot (13 x)}{- x + 5})}{(5 - x) \cdot 6 + (5 - x) (\frac{2 (3 x - 2)}{5 - x})}$

Multiply $2$ by $13$ .

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (\frac{26 x}{- x + 5})}{(5 - x) \cdot 6 + (5 - x) (\frac{2 (3 x - 2)}{5 - x})}$

Simplify the denominator.

Factor $5 - x$ out of $(5 - x) \cdot 6 + (5 - x) \frac{2 (3 x - 2)}{5 - x}$ .

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

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (\frac{26 x}{- x + 5})}{(5 - x) (6) + (5 - x) (\frac{2 (3 x - 2)}{5 - x})}$

Factor $5 - x$ out of $(5 - x) (6) + (5 - x) (\frac{2 (3 x - 2)}{5 - x})$ .

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (\frac{26 x}{- x + 5})}{(5 - x) (6 + \frac{2 (3 x - 2)}{5 - x})}$

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

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (\frac{26 x}{- x + 5})}{(5 - x) (\frac{6 (5 - x)}{5 - x} + \frac{2 (3 x - 2)}{5 - x})}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (\frac{26 x}{- x + 5})}{(5 - x) (\frac{6 (5 - x) + 2 (3 x - 2)}{5 - x})}$

Reorder terms.

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (\frac{26 x}{- x + 5})}{(5 - x) (\frac{6 (5 - x) + 2 (3 x - 2)}{- x + 5})}$

Rewrite $\frac{6 (5 - x) + 2 (3 x - 2)}{- x + 5}$ in a factored form.

Factor $2$ out of $6 (5 - x) + 2 (3 x - 2)$ .

Factor $2$ out of $6 (5 - x)$ .

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (\frac{26 x}{- x + 5})}{(5 - x) (\frac{2 (3 (5 - x)) + 2 (3 x - 2)}{- x + 5})}$

Factor $2$ out of $2 (3 (5 - x)) + 2 (3 x - 2)$ .

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (\frac{26 x}{- x + 5})}{(5 - x) (\frac{2 (3 (5 - x) + 3 x - 2)}{- x + 5})}$

Apply the distributive property.

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (\frac{26 x}{- x + 5})}{(5 - x) (\frac{2 (3 \cdot 5 + 3 (- x) + 3 x - 2)}{- x + 5})}$

Multiply $3$ by $5$ .

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (\frac{26 x}{- x + 5})}{(5 - x) (\frac{2 (15 + 3 (- x) + 3 x - 2)}{- x + 5})}$

Multiply $- 1$ by $3$ .

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (\frac{26 x}{- x + 5})}{(5 - x) (\frac{2 (15 - 3 x + 3 x - 2)}{- x + 5})}$

Subtract $2$ from $15$ .

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (\frac{26 x}{- x + 5})}{(5 - x) (\frac{2 (- 3 x + 3 x + 13)}{- x + 5})}$

Add $- 3 x$ and $3 x$ .

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (\frac{26 x}{- x + 5})}{(5 - x) (\frac{2 (0 + 13)}{- x + 5})}$

Add $0$ and $13$ .

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (\frac{26 x}{- x + 5})}{(5 - x) (\frac{2 \cdot 13}{- x + 5})}$

Multiply $2$ by $13$ .

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (\frac{26 x}{- x + 5})}{(5 - x) (\frac{26}{- x + 5})}$

Cancel the common factor of $5 - x$ .

Cancel the common factor.

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{(5 - x) (\frac{26 x}{- x + 5})}{(5 - x) (\frac{26}{- x + 5})}$

Rewrite the expression.

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{\frac{26 x}{- x + 5}}{\frac{26}{- x + 5}}$

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{26 x}{- x + 5} \cdot \frac{- x + 5}{26}$

Cancel the common factor of $26$ .

Factor $26$ out of $26 x$ .

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{26 (x)}{- x + 5} \cdot \frac{- x + 5}{26}$

Cancel the common factor.

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{26 x}{- x + 5} \cdot \frac{- x + 5}{26}$

Rewrite the expression.

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{x}{- x + 5} \cdot (- x + 5)$

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

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{x (- x + 5)}{- x + 5}$

Cancel the common factor of $- x + 5$ .

Cancel the common factor.

$f^{- 1} (\frac{6 x - 4}{5 - x}) = \frac{x (- x + 5)}{- x + 5}$

Divide $x$ by $1$ .

$f^{- 1} (\frac{6 x - 4}{5 - x}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{5 x + 4}{6 + x})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{5 x + 4}{6 + x}) = \frac{6 (\frac{5 x + 4}{6 + x}) - 4}{5 - (\frac{5 x + 4}{6 + x})}$

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

Multiply $\frac{6 \frac{5 x + 4}{6 + x} - 4}{5 - \frac{5 x + 4}{6 + x}}$ by $\frac{6 + x}{6 + x}$ .

$f (\frac{5 x + 4}{6 + x}) = \frac{6 + x}{6 + x} \cdot \frac{6 (\frac{5 x + 4}{6 + x}) - 4}{5 - \frac{5 x + 4}{6 + x}}$

Combine.

$f (\frac{5 x + 4}{6 + x}) = \frac{(6 + x) (6 (\frac{5 x + 4}{6 + x}) - 4)}{(6 + x) (5 - \frac{5 x + 4}{6 + x})}$

Apply the distributive property.

$f (\frac{5 x + 4}{6 + x}) = \frac{(6 + x) (6 (\frac{5 x + 4}{6 + x})) + (6 + x) \cdot - 4}{(6 + x) \cdot 5 + (6 + x) (- \frac{5 x + 4}{6 + x})}$

Cancel the common factor of $6 + x$ .

Move the leading negative in $- \frac{5 x + 4}{6 + x}$ into the numerator.

$f (\frac{5 x + 4}{6 + x}) = \frac{(6 + x) (6 (\frac{5 x + 4}{6 + x})) + (6 + x) \cdot - 4}{(6 + x) \cdot 5 + (6 + x) (\frac{- (5 x + 4)}{6 + x})}$

Cancel the common factor.

$f (\frac{5 x + 4}{6 + x}) = \frac{(6 + x) (6 (\frac{5 x + 4}{6 + x})) + (6 + x) \cdot - 4}{(6 + x) \cdot 5 + (6 + x) (\frac{- (5 x + 4)}{6 + x})}$

Rewrite the expression.

$f (\frac{5 x + 4}{6 + x}) = \frac{(6 + x) (6 (\frac{5 x + 4}{6 + x})) + (6 + x) \cdot - 4}{(6 + x) \cdot 5 - (5 x + 4)}$

Simplify the numerator.

Factor $(6 + x) \cdot 2$ out of $(6 + x) \cdot 6 \frac{5 x + 4}{6 + x} + (6 + x) \cdot - 4$ .

Factor $(6 + x) \cdot 2$ out of $(6 + x) \cdot 6 \frac{5 x + 4}{6 + x}$ .

$f (\frac{5 x + 4}{6 + x}) = \frac{(6 + x) \cdot (2 (3 (\frac{5 x + 4}{6 + x}))) + (6 + x) \cdot - 4}{(6 + x) \cdot 5 - (5 x + 4)}$

Factor $(6 + x) \cdot 2$ out of $(6 + x) \cdot - 4$ .

$f (\frac{5 x + 4}{6 + x}) = \frac{(6 + x) \cdot (2 (3 (\frac{5 x + 4}{6 + x}))) + (6 + x) \cdot (2 (- 2))}{(6 + x) \cdot 5 - (5 x + 4)}$

Factor $(6 + x) \cdot 2$ out of $(6 + x) \cdot 2 (3 \frac{5 x + 4}{6 + x}) + (6 + x) \cdot 2 (- 2)$ .

$f (\frac{5 x + 4}{6 + x}) = \frac{(6 + x) \cdot (2 (3 (\frac{5 x + 4}{6 + x}) - 2))}{(6 + x) \cdot 5 - (5 x + 4)}$

Combine $3$ and $\frac{5 x + 4}{6 + x}$ .

$f (\frac{5 x + 4}{6 + x}) = \frac{(6 + x) \cdot (2 (\frac{3 (5 x + 4)}{6 + x} - 2))}{(6 + x) \cdot 5 - (5 x + 4)}$

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

$f (\frac{5 x + 4}{6 + x}) = \frac{(6 + x) \cdot (2 (\frac{3 (5 x + 4)}{6 + x} - 2 \cdot \frac{6 + x}{6 + x}))}{(6 + x) \cdot 5 - (5 x + 4)}$

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

$f (\frac{5 x + 4}{6 + x}) = \frac{(6 + x) \cdot (2 (\frac{3 (5 x + 4)}{6 + x} + \frac{- 2 (6 + x)}{6 + x}))}{(6 + x) \cdot 5 - (5 x + 4)}$

Combine the numerators over the common denominator.

$f (\frac{5 x + 4}{6 + x}) = \frac{(6 + x) \cdot (2 (\frac{3 (5 x + 4) - 2 (6 + x)}{6 + x}))}{(6 + x) \cdot 5 - (5 x + 4)}$

Simplify the numerator.

Apply the distributive property.

$f (\frac{5 x + 4}{6 + x}) = \frac{(6 + x) \cdot (2 (\frac{3 (5 x) + 3 \cdot 4 - 2 (6 + x)}{6 + x}))}{(6 + x) \cdot 5 - (5 x + 4)}$

Multiply $5$ by $3$ .

$f (\frac{5 x + 4}{6 + x}) = \frac{(6 + x) \cdot (2 (\frac{15 x + 3 \cdot 4 - 2 (6 + x)}{6 + x}))}{(6 + x) \cdot 5 - (5 x + 4)}$

Multiply $3$ by $4$ .

$f (\frac{5 x + 4}{6 + x}) = \frac{(6 + x) \cdot (2 (\frac{15 x + 12 - 2 (6 + x)}{6 + x}))}{(6 + x) \cdot 5 - (5 x + 4)}$

Apply the distributive property.

$f (\frac{5 x + 4}{6 + x}) = \frac{(6 + x) \cdot (2 (\frac{15 x + 12 - 2 \cdot 6 - 2 x}{6 + x}))}{(6 + x) \cdot 5 - (5 x + 4)}$

Multiply $- 2$ by $6$ .

$f (\frac{5 x + 4}{6 + x}) = \frac{(6 + x) \cdot (2 (\frac{15 x + 12 - 12 - 2 x}{6 + x}))}{(6 + x) \cdot 5 - (5 x + 4)}$

Subtract $2 x$ from $15 x$ .

$f (\frac{5 x + 4}{6 + x}) = \frac{(6 + x) \cdot (2 (\frac{13 x + 12 - 12}{6 + x}))}{(6 + x) \cdot 5 - (5 x + 4)}$

Subtract $12$ from $12$ .

$f (\frac{5 x + 4}{6 + x}) = \frac{(6 + x) \cdot (2 (\frac{13 x + 0}{6 + x}))}{(6 + x) \cdot 5 - (5 x + 4)}$

Add $13 x$ and $0$ .

$f (\frac{5 x + 4}{6 + x}) = \frac{(6 + x) \cdot (2 (\frac{13 x}{6 + x}))}{(6 + x) \cdot 5 - (5 x + 4)}$

Combine exponents.

Combine $\frac{13 x}{6 + x}$ and $2$ .

$f (\frac{5 x + 4}{6 + x}) = \frac{(6 + x) \cdot \frac{13 x \cdot 2}{6 + x}}{(6 + x) \cdot 5 - (5 x + 4)}$

Multiply $2$ by $13$ .

$f (\frac{5 x + 4}{6 + x}) = \frac{(6 + x) \cdot \frac{26 x}{6 + x}}{(6 + x) \cdot 5 - (5 x + 4)}$

Simplify the denominator.

Apply the distributive property.

$f (\frac{5 x + 4}{6 + x}) = \frac{(6 + x) \cdot \frac{26 x}{6 + x}}{6 \cdot 5 + x \cdot 5 - (5 x + 4)}$

Multiply $6$ by $5$ .

$f (\frac{5 x + 4}{6 + x}) = \frac{(6 + x) \cdot \frac{26 x}{6 + x}}{30 + x \cdot 5 - (5 x + 4)}$

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

$f (\frac{5 x + 4}{6 + x}) = \frac{(6 + x) \cdot \frac{26 x}{6 + x}}{30 + 5 \cdot x - (5 x + 4)}$

Apply the distributive property.

$f (\frac{5 x + 4}{6 + x}) = \frac{(6 + x) \cdot \frac{26 x}{6 + x}}{30 + 5 x - (5 x) - 1 \cdot 4}$

Multiply $5$ by $- 1$ .

$f (\frac{5 x + 4}{6 + x}) = \frac{(6 + x) \cdot \frac{26 x}{6 + x}}{30 + 5 x - 5 x - 1 \cdot 4}$

Multiply $- 1$ by $4$ .

$f (\frac{5 x + 4}{6 + x}) = \frac{(6 + x) \cdot \frac{26 x}{6 + x}}{30 + 5 x - 5 x - 4}$

Subtract $4$ from $30$ .

$f (\frac{5 x + 4}{6 + x}) = \frac{(6 + x) \cdot \frac{26 x}{6 + x}}{5 x - 5 x + 26}$

Subtract $5 x$ from $5 x$ .

$f (\frac{5 x + 4}{6 + x}) = \frac{(6 + x) \cdot \frac{26 x}{6 + x}}{0 + 26}$

Add $0$ and $26$ .

$f (\frac{5 x + 4}{6 + x}) = \frac{(6 + x) \cdot \frac{26 x}{6 + x}}{26}$

Reduce the expression by cancelling the common factors.

Factor $\frac{26 x}{6 + x}$ out of $\frac{(6 + x) \cdot \frac{26 x}{6 + x}}{26}$ .

$f (\frac{5 x + 4}{6 + x}) = \frac{26 x}{6 + x} \cdot \frac{6 + x}{26}$

Cancel the common factor of $26$ .

Factor $26$ out of $26 x$ .

$f (\frac{5 x + 4}{6 + x}) = \frac{26 (x)}{6 + x} \cdot \frac{6 + x}{26}$

Cancel the common factor.

$f (\frac{5 x + 4}{6 + x}) = \frac{26 x}{6 + x} \cdot \frac{6 + x}{26}$

Rewrite the expression.

$f (\frac{5 x + 4}{6 + x}) = \frac{x}{6 + x} \cdot (6 + x)$

Cancel the common factor of $6 + x$ .

Cancel the common factor.

$f (\frac{5 x + 4}{6 + x}) = \frac{x}{6 + x} \cdot (6 + x)$

Rewrite the expression.

$f (\frac{5 x + 4}{6 + x}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{5 x + 4}{6 + x}$ is the inverse of $f (x) = \frac{6 x - 4}{5 - x}$ .

$f^{- 1} (x) = \frac{5 x + 4}{6 + x}$

Find the Inverse (6x-4)/(5-x)

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