Find the Inverse (5x-6)/(7x+8)

$\frac{5 x - 6}{7 x + 8}$

Step 1

Interchange the variables.

$x = \frac{5 y - 6}{7 y + 8}$

Step 2

Solve for $y$ .

Rewrite the equation as $\frac{5 y - 6}{7 y + 8} = x$ .

$\frac{5 y - 6}{7 y + 8} = x$

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.

$7 y + 8, 1$

Remove parentheses.

$7 y + 8, 1$

The LCM of one and any expression is the expression.

$7 y + 8$

Multiply each term in $\frac{5 y - 6}{7 y + 8} = x$ by $7 y + 8$ to eliminate the fractions.

Multiply each term in $\frac{5 y - 6}{7 y + 8} = x$ by $7 y + 8$ .

$\frac{5 y - 6}{7 y + 8} (7 y + 8) = x (7 y + 8)$

Simplify the left side.

Cancel the common factor of $7 y + 8$ .

Cancel the common factor.

$\frac{5 y - 6}{7 y + 8} (7 y + 8) = x (7 y + 8)$

Rewrite the expression.

$5 y - 6 = x (7 y + 8)$

Simplify the right side.

Apply the distributive property.

$5 y - 6 = x (7 y) + x \cdot 8$

Reorder.

Rewrite using the commutative property of multiplication.

$5 y - 6 = 7 x y + x \cdot 8$

Move $8$ to the left of $x$ .

$5 y - 6 = 7 x y + 8 x$

Solve the equation.

Subtract $7 x y$ from both sides of the equation.

$5 y - 6 - 7 x y = 8 x$

Add $6$ to both sides of the equation.

$5 y - 7 x y = 8 x + 6$

Factor $y$ out of $5 y - 7 x y$ .

Factor $y$ out of $5 y$ .

$y \cdot 5 - 7 x y = 8 x + 6$

Factor $y$ out of $- 7 x y$ .

$y \cdot 5 + y (- 7 x) = 8 x + 6$

Factor $y$ out of $y \cdot 5 + y (- 7 x)$ .

$y (5 - 7 x) = 8 x + 6$

Divide each term in $y (5 - 7 x) = 8 x + 6$ by $5 - 7 x$ and simplify.

Divide each term in $y (5 - 7 x) = 8 x + 6$ by $5 - 7 x$ .

$\frac{y (5 - 7 x)}{5 - 7 x} = \frac{8 x}{5 - 7 x} + \frac{6}{5 - 7 x}$

Simplify the left side.

Cancel the common factor of $5 - 7 x$ .

Cancel the common factor.

$\frac{y (5 - 7 x)}{5 - 7 x} = \frac{8 x}{5 - 7 x} + \frac{6}{5 - 7 x}$

Divide $y$ by $1$ .

$y = \frac{8 x}{5 - 7 x} + \frac{6}{5 - 7 x}$

Simplify the right side.

Combine the numerators over the common denominator.

$y = \frac{8 x + 6}{5 - 7 x}$

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

Factor $2$ out of $8 x$ .

$y = \frac{2 (4 x) + 6}{5 - 7 x}$

Factor $2$ out of $6$ .

$y = \frac{2 (4 x) + 2 (3)}{5 - 7 x}$

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

$y = \frac{2 (4 x + 3)}{5 - 7 x}$

Step 3

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

$f^{- 1} (x) = \frac{2 (4 x + 3)}{5 - 7 x}$

Step 4

Verify if $f^{- 1} (x) = \frac{2 (4 x + 3)}{5 - 7 x}$ is the inverse of $f (x) = \frac{5 x - 6}{7 x + 8}$ .

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 $f^{- 1} (\frac{5 x - 6}{7 x + 8})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{5 x - 6}{7 x + 8}) = \frac{2 (4 (\frac{5 x - 6}{7 x + 8}) + 3)}{5 - 7 \frac{5 x - 6}{7 x + 8}}$

Simplify the numerator.

Combine $4$ and $\frac{5 x - 6}{7 x + 8}$ .

$f^{- 1} (\frac{5 x - 6}{7 x + 8}) = \frac{2 (\frac{4 (5 x - 6)}{7 x + 8} + 3)}{5 - 7 \frac{5 x - 6}{7 x + 8}}$

To write $3$ as a fraction with a common denominator, multiply by $\frac{7 x + 8}{7 x + 8}$ .

$f^{- 1} (\frac{5 x - 6}{7 x + 8}) = \frac{2 (\frac{4 (5 x - 6)}{7 x + 8} + \frac{3 (7 x + 8)}{7 x + 8})}{5 - 7 \frac{5 x - 6}{7 x + 8}}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{5 x - 6}{7 x + 8}) = \frac{2 (\frac{4 (5 x - 6) + 3 (7 x + 8)}{7 x + 8})}{5 - 7 \frac{5 x - 6}{7 x + 8}}$

Rewrite $\frac{4 (5 x - 6) + 3 (7 x + 8)}{7 x + 8}$ in a factored form.

Apply the distributive property.

$f^{- 1} (\frac{5 x - 6}{7 x + 8}) = \frac{2 (\frac{4 (5 x) + 4 \cdot - 6 + 3 (7 x + 8)}{7 x + 8})}{5 - 7 \frac{5 x - 6}{7 x + 8}}$

Multiply $5$ by $4$ .

$f^{- 1} (\frac{5 x - 6}{7 x + 8}) = \frac{2 (\frac{20 x + 4 \cdot - 6 + 3 (7 x + 8)}{7 x + 8})}{5 - 7 \frac{5 x - 6}{7 x + 8}}$

Multiply $4$ by $- 6$ .

$f^{- 1} (\frac{5 x - 6}{7 x + 8}) = \frac{2 (\frac{20 x - 24 + 3 (7 x + 8)}{7 x + 8})}{5 - 7 \frac{5 x - 6}{7 x + 8}}$

Apply the distributive property.

$f^{- 1} (\frac{5 x - 6}{7 x + 8}) = \frac{2 (\frac{20 x - 24 + 3 (7 x) + 3 \cdot 8}{7 x + 8})}{5 - 7 \frac{5 x - 6}{7 x + 8}}$

Multiply $7$ by $3$ .

$f^{- 1} (\frac{5 x - 6}{7 x + 8}) = \frac{2 (\frac{20 x - 24 + 21 x + 3 \cdot 8}{7 x + 8})}{5 - 7 \frac{5 x - 6}{7 x + 8}}$

Multiply $3$ by $8$ .

$f^{- 1} (\frac{5 x - 6}{7 x + 8}) = \frac{2 (\frac{20 x - 24 + 21 x + 24}{7 x + 8})}{5 - 7 \frac{5 x - 6}{7 x + 8}}$

Add $20 x$ and $21 x$ .

$f^{- 1} (\frac{5 x - 6}{7 x + 8}) = \frac{2 (\frac{41 x - 24 + 24}{7 x + 8})}{5 - 7 \frac{5 x - 6}{7 x + 8}}$

Add $- 24$ and $24$ .

$f^{- 1} (\frac{5 x - 6}{7 x + 8}) = \frac{2 (\frac{41 x + 0}{7 x + 8})}{5 - 7 \frac{5 x - 6}{7 x + 8}}$

Add $41 x$ and $0$ .

$f^{- 1} (\frac{5 x - 6}{7 x + 8}) = \frac{2 (\frac{41 x}{7 x + 8})}{5 - 7 \frac{5 x - 6}{7 x + 8}}$

Simplify the denominator.

Combine $- 7$ and $\frac{5 x - 6}{7 x + 8}$ .

$f^{- 1} (\frac{5 x - 6}{7 x + 8}) = \frac{2 (\frac{41 x}{7 x + 8})}{5 + \frac{- 7 (5 x - 6)}{7 x + 8}}$

Move the negative in front of the fraction.

$f^{- 1} (\frac{5 x - 6}{7 x + 8}) = \frac{2 (\frac{41 x}{7 x + 8})}{5 - \frac{(7) (5 x - 6)}{7 x + 8}}$

To write $5$ as a fraction with a common denominator, multiply by $\frac{7 x + 8}{7 x + 8}$ .

$f^{- 1} (\frac{5 x - 6}{7 x + 8}) = \frac{2 (\frac{41 x}{7 x + 8})}{\frac{5 (7 x + 8)}{7 x + 8} - \frac{7 (5 x - 6)}{7 x + 8}}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{5 x - 6}{7 x + 8}) = \frac{2 (\frac{41 x}{7 x + 8})}{\frac{5 (7 x + 8) - 7 (5 x - 6)}{7 x + 8}}$

Rewrite $\frac{5 (7 x + 8) - 7 (5 x - 6)}{7 x + 8}$ in a factored form.

Apply the distributive property.

$f^{- 1} (\frac{5 x - 6}{7 x + 8}) = \frac{2 (\frac{41 x}{7 x + 8})}{\frac{5 (7 x) + 5 \cdot 8 - 7 (5 x - 6)}{7 x + 8}}$

Multiply $7$ by $5$ .

$f^{- 1} (\frac{5 x - 6}{7 x + 8}) = \frac{2 (\frac{41 x}{7 x + 8})}{\frac{35 x + 5 \cdot 8 - 7 (5 x - 6)}{7 x + 8}}$

Multiply $5$ by $8$ .

$f^{- 1} (\frac{5 x - 6}{7 x + 8}) = \frac{2 (\frac{41 x}{7 x + 8})}{\frac{35 x + 40 - 7 (5 x - 6)}{7 x + 8}}$

Apply the distributive property.

$f^{- 1} (\frac{5 x - 6}{7 x + 8}) = \frac{2 (\frac{41 x}{7 x + 8})}{\frac{35 x + 40 - 7 (5 x) - 7 \cdot - 6}{7 x + 8}}$

Multiply $5$ by $- 7$ .

$f^{- 1} (\frac{5 x - 6}{7 x + 8}) = \frac{2 (\frac{41 x}{7 x + 8})}{\frac{35 x + 40 - 35 x - 7 \cdot - 6}{7 x + 8}}$

Multiply $- 7$ by $- 6$ .

$f^{- 1} (\frac{5 x - 6}{7 x + 8}) = \frac{2 (\frac{41 x}{7 x + 8})}{\frac{35 x + 40 - 35 x + 42}{7 x + 8}}$

Subtract $35 x$ from $35 x$ .

$f^{- 1} (\frac{5 x - 6}{7 x + 8}) = \frac{2 (\frac{41 x}{7 x + 8})}{\frac{0 + 40 + 42}{7 x + 8}}$

Add $0$ and $40$ .

$f^{- 1} (\frac{5 x - 6}{7 x + 8}) = \frac{2 (\frac{41 x}{7 x + 8})}{\frac{40 + 42}{7 x + 8}}$

Add $40$ and $42$ .

$f^{- 1} (\frac{5 x - 6}{7 x + 8}) = \frac{2 (\frac{41 x}{7 x + 8})}{\frac{82}{7 x + 8}}$

Combine $2$ and $\frac{41 x}{7 x + 8}$ .

$f^{- 1} (\frac{5 x - 6}{7 x + 8}) = \frac{\frac{2 (41 x)}{7 x + 8}}{\frac{82}{7 x + 8}}$

Multiply $2$ by $41$ .

$f^{- 1} (\frac{5 x - 6}{7 x + 8}) = \frac{\frac{82 x}{7 x + 8}}{\frac{82}{7 x + 8}}$

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{5 x - 6}{7 x + 8}) = \frac{82 x}{7 x + 8} \cdot \frac{7 x + 8}{82}$

Cancel the common factor of $82$ .

Factor $82$ out of $82 x$ .

$f^{- 1} (\frac{5 x - 6}{7 x + 8}) = \frac{82 (x)}{7 x + 8} \cdot \frac{7 x + 8}{82}$

Cancel the common factor.

$f^{- 1} (\frac{5 x - 6}{7 x + 8}) = \frac{82 x}{7 x + 8} \cdot \frac{7 x + 8}{82}$

Rewrite the expression.

$f^{- 1} (\frac{5 x - 6}{7 x + 8}) = \frac{x}{7 x + 8} \cdot (7 x + 8)$

Cancel the common factor of $7 x + 8$ .

Cancel the common factor.

$f^{- 1} (\frac{5 x - 6}{7 x + 8}) = \frac{x}{7 x + 8} \cdot (7 x + 8)$

Rewrite the expression.

$f^{- 1} (\frac{5 x - 6}{7 x + 8}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{2 (4 x + 3)}{5 - 7 x})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{5 (\frac{2 (4 x + 3)}{5 - 7 x}) - 6}{7 (\frac{2 (4 x + 3)}{5 - 7 x}) + 8}$

Simplify the numerator.

Multiply $5 \frac{2 (4 x + 3)}{5 - 7 x}$ .

Combine $5$ and $\frac{2 (4 x + 3)}{5 - 7 x}$ .

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{5 (2 (4 x + 3))}{5 - 7 x} - 6}{7 (\frac{2 (4 x + 3)}{5 - 7 x}) + 8}$

Multiply $2$ by $5$ .

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{10 (4 x + 3)}{5 - 7 x} - 6}{7 (\frac{2 (4 x + 3)}{5 - 7 x}) + 8}$

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{5 - 7 x}{5 - 7 x}$ .

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{10 (4 x + 3)}{5 - 7 x} - 6 \cdot \frac{5 - 7 x}{5 - 7 x}}{7 (\frac{2 (4 x + 3)}{5 - 7 x}) + 8}$

Combine $- 6$ and $\frac{5 - 7 x}{5 - 7 x}$ .

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{10 (4 x + 3)}{5 - 7 x} + \frac{- 6 (5 - 7 x)}{5 - 7 x}}{7 (\frac{2 (4 x + 3)}{5 - 7 x}) + 8}$

Combine the numerators over the common denominator.

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{10 (4 x + 3) - 6 (5 - 7 x)}{5 - 7 x}}{7 (\frac{2 (4 x + 3)}{5 - 7 x}) + 8}$

Reorder terms.

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{10 (4 x + 3) - 6 (5 - 7 x)}{- 7 x + 5}}{7 (\frac{2 (4 x + 3)}{5 - 7 x}) + 8}$

Rewrite $\frac{10 (4 x + 3) - 6 (5 - 7 x)}{- 7 x + 5}$ in a factored form.

Factor $2$ out of $10 (4 x + 3) - 6 (5 - 7 x)$ .

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

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{2 (5 (4 x + 3)) - 6 (5 - 7 x)}{- 7 x + 5}}{7 (\frac{2 (4 x + 3)}{5 - 7 x}) + 8}$

Factor $2$ out of $- 6 (5 - 7 x)$ .

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{2 (5 (4 x + 3)) + 2 (- 3 (5 - 7 x))}{- 7 x + 5}}{7 (\frac{2 (4 x + 3)}{5 - 7 x}) + 8}$

Factor $2$ out of $2 (5 (4 x + 3)) + 2 (- 3 (5 - 7 x))$ .

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{2 (5 (4 x + 3) - 3 (5 - 7 x))}{- 7 x + 5}}{7 (\frac{2 (4 x + 3)}{5 - 7 x}) + 8}$

Apply the distributive property.

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{2 (5 (4 x) + 5 \cdot 3 - 3 (5 - 7 x))}{- 7 x + 5}}{7 (\frac{2 (4 x + 3)}{5 - 7 x}) + 8}$

Multiply $4$ by $5$ .

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{2 (20 x + 5 \cdot 3 - 3 (5 - 7 x))}{- 7 x + 5}}{7 (\frac{2 (4 x + 3)}{5 - 7 x}) + 8}$

Multiply $5$ by $3$ .

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{2 (20 x + 15 - 3 (5 - 7 x))}{- 7 x + 5}}{7 (\frac{2 (4 x + 3)}{5 - 7 x}) + 8}$

Apply the distributive property.

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{2 (20 x + 15 - 3 \cdot 5 - 3 (- 7 x))}{- 7 x + 5}}{7 (\frac{2 (4 x + 3)}{5 - 7 x}) + 8}$

Multiply $- 3$ by $5$ .

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{2 (20 x + 15 - 15 - 3 (- 7 x))}{- 7 x + 5}}{7 (\frac{2 (4 x + 3)}{5 - 7 x}) + 8}$

Multiply $- 7$ by $- 3$ .

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{2 (20 x + 15 - 15 + 21 x)}{- 7 x + 5}}{7 (\frac{2 (4 x + 3)}{5 - 7 x}) + 8}$

Add $20 x$ and $21 x$ .

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{2 (41 x + 15 - 15)}{- 7 x + 5}}{7 (\frac{2 (4 x + 3)}{5 - 7 x}) + 8}$

Subtract $15$ from $15$ .

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{2 (41 x + 0)}{- 7 x + 5}}{7 (\frac{2 (4 x + 3)}{5 - 7 x}) + 8}$

Add $41 x$ and $0$ .

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{2 \cdot (41 x)}{- 7 x + 5}}{7 (\frac{2 (4 x + 3)}{5 - 7 x}) + 8}$

Multiply $2$ by $41$ .

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{82 x}{- 7 x + 5}}{7 (\frac{2 (4 x + 3)}{5 - 7 x}) + 8}$

Simplify the denominator.

Multiply $7 \frac{2 (4 x + 3)}{5 - 7 x}$ .

Combine $7$ and $\frac{2 (4 x + 3)}{5 - 7 x}$ .

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{82 x}{- 7 x + 5}}{\frac{7 (2 (4 x + 3))}{5 - 7 x} + 8}$

Multiply $2$ by $7$ .

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{82 x}{- 7 x + 5}}{\frac{14 (4 x + 3)}{5 - 7 x} + 8}$

To write $8$ as a fraction with a common denominator, multiply by $\frac{5 - 7 x}{5 - 7 x}$ .

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{82 x}{- 7 x + 5}}{\frac{14 (4 x + 3)}{5 - 7 x} + \frac{8 (5 - 7 x)}{5 - 7 x}}$

Combine the numerators over the common denominator.

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{82 x}{- 7 x + 5}}{\frac{14 (4 x + 3) + 8 (5 - 7 x)}{5 - 7 x}}$

Reorder terms.

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{82 x}{- 7 x + 5}}{\frac{14 (4 x + 3) + 8 (5 - 7 x)}{- 7 x + 5}}$

Rewrite $\frac{14 (4 x + 3) + 8 (5 - 7 x)}{- 7 x + 5}$ in a factored form.

Factor $2$ out of $14 (4 x + 3) + 8 (5 - 7 x)$ .

Factor $2$ out of $14 (4 x + 3)$ .

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{82 x}{- 7 x + 5}}{\frac{2 (7 (4 x + 3)) + 8 (5 - 7 x)}{- 7 x + 5}}$

Factor $2$ out of $8 (5 - 7 x)$ .

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{82 x}{- 7 x + 5}}{\frac{2 (7 (4 x + 3)) + 2 (4 (5 - 7 x))}{- 7 x + 5}}$

Factor $2$ out of $2 (7 (4 x + 3)) + 2 (4 (5 - 7 x))$ .

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{82 x}{- 7 x + 5}}{\frac{2 (7 (4 x + 3) + 4 (5 - 7 x))}{- 7 x + 5}}$

Apply the distributive property.

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{82 x}{- 7 x + 5}}{\frac{2 (7 (4 x) + 7 \cdot 3 + 4 (5 - 7 x))}{- 7 x + 5}}$

Multiply $4$ by $7$ .

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{82 x}{- 7 x + 5}}{\frac{2 (28 x + 7 \cdot 3 + 4 (5 - 7 x))}{- 7 x + 5}}$

Multiply $7$ by $3$ .

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{82 x}{- 7 x + 5}}{\frac{2 (28 x + 21 + 4 (5 - 7 x))}{- 7 x + 5}}$

Apply the distributive property.

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{82 x}{- 7 x + 5}}{\frac{2 (28 x + 21 + 4 \cdot 5 + 4 (- 7 x))}{- 7 x + 5}}$

Multiply $4$ by $5$ .

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{82 x}{- 7 x + 5}}{\frac{2 (28 x + 21 + 20 + 4 (- 7 x))}{- 7 x + 5}}$

Multiply $- 7$ by $4$ .

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{82 x}{- 7 x + 5}}{\frac{2 (28 x + 21 + 20 - 28 x)}{- 7 x + 5}}$

Subtract $28 x$ from $28 x$ .

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{82 x}{- 7 x + 5}}{\frac{2 (0 + 21 + 20)}{- 7 x + 5}}$

Add $0$ and $21$ .

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{82 x}{- 7 x + 5}}{\frac{2 (21 + 20)}{- 7 x + 5}}$

Add $21$ and $20$ .

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{82 x}{- 7 x + 5}}{\frac{2 \cdot 41}{- 7 x + 5}}$

Multiply $2$ by $41$ .

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{\frac{82 x}{- 7 x + 5}}{\frac{82}{- 7 x + 5}}$

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{82 x}{- 7 x + 5} \cdot \frac{- 7 x + 5}{82}$

Cancel the common factor of $82$ .

Factor $82$ out of $82 x$ .

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{82 (x)}{- 7 x + 5} \cdot \frac{- 7 x + 5}{82}$

Cancel the common factor.

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{82 x}{- 7 x + 5} \cdot \frac{- 7 x + 5}{82}$

Rewrite the expression.

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{x}{- 7 x + 5} \cdot (- 7 x + 5)$

Multiply $\frac{x}{- 7 x + 5}$ by $- 7 x + 5$ .

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{x (- 7 x + 5)}{- 7 x + 5}$

Cancel the common factor of $- 7 x + 5$ .

Cancel the common factor.

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = \frac{x (- 7 x + 5)}{- 7 x + 5}$

Divide $x$ by $1$ .

$f (\frac{2 (4 x + 3)}{5 - 7 x}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{2 (4 x + 3)}{5 - 7 x}$ is the inverse of $f (x) = \frac{5 x - 6}{7 x + 8}$ .

$f^{- 1} (x) = \frac{2 (4 x + 3)}{5 - 7 x}$

Find the Inverse (5x-6)/(7x+8)

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