Find the Inverse (-6x-3)/(-10x+8)

$\frac{- 6 x - 3}{- 10 x + 8}$

Step 1

Interchange the variables.

$x = \frac{- 6 y - 3}{- 10 y + 8}$

Step 2

Solve for $y$ .

Rewrite the equation as $\frac{- 6 y - 3}{- 10 y + 8} = x$ .

$\frac{- 6 y - 3}{- 10 y + 8} = x$

Factor each term.

Factor $3$ out of $- 6 y - 3$ .

Factor $3$ out of $- 6 y$ .

$\frac{3 (- 2 y) - 3}{- 10 y + 8} = x$

Factor $3$ out of $- 3$ .

$\frac{3 (- 2 y) + 3 (- 1)}{- 10 y + 8} = x$

Factor $3$ out of $3 (- 2 y) + 3 (- 1)$ .

$\frac{3 (- 2 y - 1)}{- 10 y + 8} = x$

Factor $2$ out of $- 10 y + 8$ .

Factor $2$ out of $- 10 y$ .

$\frac{3 (- 2 y - 1)}{2 (- 5 y) + 8} = x$

Factor $2$ out of $8$ .

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

Factor $2$ out of $2 (- 5 y) + 2 (4)$ .

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

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.

$2 (- 5 y + 4), 1$

The LCM of one and any expression is the expression.

$2 (- 5 y + 4)$

Multiply each term in $\frac{3 (- 2 y - 1)}{2 (- 5 y + 4)} = x$ by $2 (- 5 y + 4)$ to eliminate the fractions.

Multiply each term in $\frac{3 (- 2 y - 1)}{2 (- 5 y + 4)} = x$ by $2 (- 5 y + 4)$ .

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

Simplify the left side.

Rewrite using the commutative property of multiplication.

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $\frac{3 (- 2 y - 1)}{- 5 y + 4}$ by $- 5 y + 4$ .

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

Cancel the common factor of $- 5 y + 4$ .

Cancel the common factor.

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

Divide $3 (- 2 y - 1)$ by $1$ .

$3 (- 2 y - 1) = x (2 (- 5 y + 4))$

Apply the distributive property.

$3 (- 2 y) + 3 \cdot - 1 = x (2 (- 5 y + 4))$

Multiply.

Multiply $- 2$ by $3$ .

$- 6 y + 3 \cdot - 1 = x (2 (- 5 y + 4))$

Multiply $3$ by $- 1$ .

$- 6 y - 3 = x (2 (- 5 y + 4))$

Simplify the right side.

Rewrite using the commutative property of multiplication.

$- 6 y - 3 = 2 x (- 5 y + 4)$

Apply the distributive property.

$- 6 y - 3 = 2 x (- 5 y) + 2 x \cdot 4$

Simplify the expression.

Rewrite using the commutative property of multiplication.

$- 6 y - 3 = 2 \cdot - 5 x y + 2 x \cdot 4$

Multiply $4$ by $2$ .

$- 6 y - 3 = 2 \cdot - 5 x y + 8 x$

Multiply $2$ by $- 5$ .

$- 6 y - 3 = - 10 x y + 8 x$

Solve the equation.

Add $10 x y$ to both sides of the equation.

$- 6 y - 3 + 10 x y = 8 x$

Add $3$ to both sides of the equation.

$- 6 y + 10 x y = 8 x + 3$

Factor $2 y$ out of $- 6 y + 10 x y$ .

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

$2 y (- 3) + 10 x y = 8 x + 3$

Factor $2 y$ out of $10 x y$ .

$2 y (- 3) + 2 y (5 x) = 8 x + 3$

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

$2 y (- 3 + 5 x) = 8 x + 3$

Divide each term in $2 y (- 3 + 5 x) = 8 x + 3$ by $2 (- 3 + 5 x)$ and simplify.

Divide each term in $2 y (- 3 + 5 x) = 8 x + 3$ by $2 (- 3 + 5 x)$ .

$\frac{2 y (- 3 + 5 x)}{2 (- 3 + 5 x)} = \frac{8 x}{2 (- 3 + 5 x)} + \frac{3}{2 (- 3 + 5 x)}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 y (- 3 + 5 x)}{2 (- 3 + 5 x)} = \frac{8 x}{2 (- 3 + 5 x)} + \frac{3}{2 (- 3 + 5 x)}$

Rewrite the expression.

$\frac{y (- 3 + 5 x)}{- 3 + 5 x} = \frac{8 x}{2 (- 3 + 5 x)} + \frac{3}{2 (- 3 + 5 x)}$

Cancel the common factor of $- 3 + 5 x$ .

Cancel the common factor.

$\frac{y (- 3 + 5 x)}{- 3 + 5 x} = \frac{8 x}{2 (- 3 + 5 x)} + \frac{3}{2 (- 3 + 5 x)}$

Divide $y$ by $1$ .

$y = \frac{8 x}{2 (- 3 + 5 x)} + \frac{3}{2 (- 3 + 5 x)}$

Simplify the right side.

Cancel the common factor of $8$ and $2$ .

Factor $2$ out of $8 x$ .

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

Cancel the common factors.

Cancel the common factor.

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

Rewrite the expression.

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

To write $\frac{4 x}{- 3 + 5 x}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Write each expression with a common denominator of $(- 3 + 5 x) \cdot 2$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{4 x}{- 3 + 5 x}$ by $\frac{2}{2}$ .

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

Reorder the factors of $(- 3 + 5 x) \cdot 2$ .

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

Combine the numerators over the common denominator.

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

Multiply $2$ by $4$ .

$y = \frac{8 x + 3}{2 (- 3 + 5 x)}$

Rewrite $- 3$ as $- 1 (3)$ .

$y = \frac{8 x + 3}{2 (- 1 (3) + 5 x)}$

Factor $- 1$ out of $5 x$ .

$y = \frac{8 x + 3}{2 (- 1 (3) - (- 5 x))}$

Factor $- 1$ out of $- 1 (3) - (- 5 x)$ .

$y = \frac{8 x + 3}{2 (- 1 (3 - 5 x))}$

Move the negative in front of the fraction.

$y = - \frac{8 x + 3}{2 (3 - 5 x)}$

Step 3

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

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

Step 4

Verify if $f^{- 1} (x) = - \frac{8 x + 3}{2 (3 - 5 x)}$ is the inverse of $f (x) = \frac{- 6 x - 3}{- 10 x + 8}$ .

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 $f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{8 (\frac{- 6 x - 3}{- 10 x + 8}) + 3}{2 (3 - 5 \frac{- 6 x - 3}{- 10 x + 8})}$

Simplify with factoring out.

Factor $3$ out of $- 6 x - 3$ .

Factor $3$ out of $- 6 x$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{8 (\frac{3 (- 2 x) - 3}{- 10 x + 8}) + 3}{2 (3 - 5 \frac{- 6 x - 3}{- 10 x + 8})}$

Factor $3$ out of $- 3$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{8 (\frac{3 (- 2 x) + 3 (- 1)}{- 10 x + 8}) + 3}{2 (3 - 5 \frac{- 6 x - 3}{- 10 x + 8})}$

Factor $3$ out of $3 (- 2 x) + 3 (- 1)$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{8 (\frac{3 (- 2 x - 1)}{- 10 x + 8}) + 3}{2 (3 - 5 \frac{- 6 x - 3}{- 10 x + 8})}$

Factor $2$ out of $- 10 x + 8$ .

Factor $2$ out of $- 10 x$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{8 (\frac{3 (- 2 x - 1)}{2 (- 5 x) + 8}) + 3}{2 (3 - 5 \frac{- 6 x - 3}{- 10 x + 8})}$

Factor $2$ out of $8$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{8 (\frac{3 (- 2 x - 1)}{2 (- 5 x) + 2 (4)}) + 3}{2 (3 - 5 \frac{- 6 x - 3}{- 10 x + 8})}$

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

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{8 (\frac{3 (- 2 x - 1)}{2 (- 5 x + 4)}) + 3}{2 (3 - 5 \frac{- 6 x - 3}{- 10 x + 8})}$

Factor $3$ out of $- 6 x - 3$ .

Factor $3$ out of $- 6 x$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{8 (\frac{3 (- 2 x - 1)}{2 (- 5 x + 4)}) + 3}{2 (3 - 5 \frac{3 (- 2 x) - 3}{- 10 x + 8})}$

Factor $3$ out of $- 3$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{8 (\frac{3 (- 2 x - 1)}{2 (- 5 x + 4)}) + 3}{2 (3 - 5 \frac{3 (- 2 x) + 3 (- 1)}{- 10 x + 8})}$

Factor $3$ out of $3 (- 2 x) + 3 (- 1)$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{8 (\frac{3 (- 2 x - 1)}{2 (- 5 x + 4)}) + 3}{2 (3 - 5 \frac{3 (- 2 x - 1)}{- 10 x + 8})}$

Factor $2$ out of $- 10 x + 8$ .

Factor $2$ out of $- 10 x$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{8 (\frac{3 (- 2 x - 1)}{2 (- 5 x + 4)}) + 3}{2 (3 - 5 \frac{3 (- 2 x - 1)}{2 (- 5 x) + 8})}$

Factor $2$ out of $8$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{8 (\frac{3 (- 2 x - 1)}{2 (- 5 x + 4)}) + 3}{2 (3 - 5 \frac{3 (- 2 x - 1)}{2 (- 5 x) + 2 (4)})}$

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

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{8 (\frac{3 (- 2 x - 1)}{2 (- 5 x + 4)}) + 3}{2 (3 - 5 \frac{3 (- 2 x - 1)}{2 (- 5 x + 4)})}$

Simplify the numerator.

Cancel the common factor of $2$ .

Factor $2$ out of $8$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{2 (4) (\frac{3 (- 2 x - 1)}{2 (- 5 x + 4)}) + 3}{2 (3 - 5 \frac{3 (- 2 x - 1)}{2 (- 5 x + 4)})}$

Cancel the common factor.

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{2 \cdot (4 (\frac{3 (- 2 x - 1)}{2 (- 5 x + 4)})) + 3}{2 (3 - 5 \frac{3 (- 2 x - 1)}{2 (- 5 x + 4)})}$

Rewrite the expression.

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{4 (\frac{3 (- 2 x - 1)}{- 5 x + 4}) + 3}{2 (3 - 5 \frac{3 (- 2 x - 1)}{2 (- 5 x + 4)})}$

Combine $4$ and $\frac{3 (- 2 x - 1)}{- 5 x + 4}$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{\frac{4 (3 (- 2 x - 1))}{- 5 x + 4} + 3}{2 (3 - 5 \frac{3 (- 2 x - 1)}{2 (- 5 x + 4)})}$

Multiply $3$ by $4$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{\frac{12 (- 2 x - 1)}{- 5 x + 4} + 3}{2 (3 - 5 \frac{3 (- 2 x - 1)}{2 (- 5 x + 4)})}$

To write $3$ as a fraction with a common denominator, multiply by $\frac{- 5 x + 4}{- 5 x + 4}$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{\frac{12 (- 2 x - 1)}{- 5 x + 4} + \frac{3 (- 5 x + 4)}{- 5 x + 4}}{2 (3 - 5 \frac{3 (- 2 x - 1)}{2 (- 5 x + 4)})}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{\frac{12 (- 2 x - 1) + 3 (- 5 x + 4)}{- 5 x + 4}}{2 (3 - 5 \frac{3 (- 2 x - 1)}{2 (- 5 x + 4)})}$

Rewrite $\frac{12 (- 2 x - 1) + 3 (- 5 x + 4)}{- 5 x + 4}$ in a factored form.

Factor $3$ out of $12 (- 2 x - 1) + 3 (- 5 x + 4)$ .

Factor $3$ out of $12 (- 2 x - 1)$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{\frac{3 (4 (- 2 x - 1)) + 3 (- 5 x + 4)}{- 5 x + 4}}{2 (3 - 5 \frac{3 (- 2 x - 1)}{2 (- 5 x + 4)})}$

Factor $3$ out of $3 (4 (- 2 x - 1)) + 3 (- 5 x + 4)$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{\frac{3 (4 (- 2 x - 1) - 5 x + 4)}{- 5 x + 4}}{2 (3 - 5 \frac{3 (- 2 x - 1)}{2 (- 5 x + 4)})}$

Apply the distributive property.

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{\frac{3 (4 (- 2 x) + 4 \cdot - 1 - 5 x + 4)}{- 5 x + 4}}{2 (3 - 5 \frac{3 (- 2 x - 1)}{2 (- 5 x + 4)})}$

Multiply $- 2$ by $4$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{\frac{3 (- 8 x + 4 \cdot - 1 - 5 x + 4)}{- 5 x + 4}}{2 (3 - 5 \frac{3 (- 2 x - 1)}{2 (- 5 x + 4)})}$

Multiply $4$ by $- 1$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{\frac{3 (- 8 x - 4 - 5 x + 4)}{- 5 x + 4}}{2 (3 - 5 \frac{3 (- 2 x - 1)}{2 (- 5 x + 4)})}$

Subtract $5 x$ from $- 8 x$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{\frac{3 (- 13 x - 4 + 4)}{- 5 x + 4}}{2 (3 - 5 \frac{3 (- 2 x - 1)}{2 (- 5 x + 4)})}$

Add $- 4$ and $4$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{\frac{3 (- 13 x + 0)}{- 5 x + 4}}{2 (3 - 5 \frac{3 (- 2 x - 1)}{2 (- 5 x + 4)})}$

Add $- 13 x$ and $0$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{\frac{3 \cdot (- 13 x)}{- 5 x + 4}}{2 (3 - 5 \frac{3 (- 2 x - 1)}{2 (- 5 x + 4)})}$

Multiply $3$ by $- 13$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{\frac{- 39 x}{- 5 x + 4}}{2 (3 - 5 \frac{3 (- 2 x - 1)}{2 (- 5 x + 4)})}$

Move the negative in front of the fraction.

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{- \frac{39 x}{- 5 x + 4}}{2 (3 - 5 \frac{3 (- 2 x - 1)}{2 (- 5 x + 4)})}$

Simplify the denominator.

Multiply $- 5 \frac{3 (- 2 x - 1)}{2 (- 5 x + 4)}$ .

Combine $- 5$ and $\frac{3 (- 2 x - 1)}{2 (- 5 x + 4)}$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{- \frac{39 x}{- 5 x + 4}}{2 (3 + \frac{- 5 (3 (- 2 x - 1))}{2 (- 5 x + 4)})}$

Multiply $3$ by $- 5$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{- \frac{39 x}{- 5 x + 4}}{2 (3 + \frac{- 15 (- 2 x - 1)}{2 (- 5 x + 4)})}$

Move the negative in front of the fraction.

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{- \frac{39 x}{- 5 x + 4}}{2 (3 - \frac{(15) (- 2 x - 1)}{2 (- 5 x + 4)})}$

To write $3$ as a fraction with a common denominator, multiply by $\frac{2 (- 5 x + 4)}{2 (- 5 x + 4)}$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{- \frac{39 x}{- 5 x + 4}}{2 (3 \cdot \frac{2 (- 5 x + 4)}{2 (- 5 x + 4)} - \frac{15 (- 2 x - 1)}{2 (- 5 x + 4)})}$

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

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{- \frac{39 x}{- 5 x + 4}}{2 (\frac{3 (2 (- 5 x + 4))}{2 (- 5 x + 4)} - \frac{15 (- 2 x - 1)}{2 (- 5 x + 4)})}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{- \frac{39 x}{- 5 x + 4}}{2 (\frac{3 (2 (- 5 x + 4)) - 15 (- 2 x - 1)}{2 (- 5 x + 4)})}$

Rewrite $\frac{3 (2 (- 5 x + 4)) - 15 (- 2 x - 1)}{2 (- 5 x + 4)}$ in a factored form.

Factor $3$ out of $3 \cdot 2 (- 5 x + 4) - 15 (- 2 x - 1)$ .

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

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{- \frac{39 x}{- 5 x + 4}}{2 (\frac{3 (2 (- 5 x + 4)) - 15 (- 2 x - 1)}{2 (- 5 x + 4)})}$

Factor $3$ out of $- 15 (- 2 x - 1)$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{- \frac{39 x}{- 5 x + 4}}{2 (\frac{3 (2 (- 5 x + 4)) + 3 (- 5 (- 2 x - 1))}{2 (- 5 x + 4)})}$

Factor $3$ out of $3 (2 (- 5 x + 4)) + 3 (- 5 (- 2 x - 1))$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{- \frac{39 x}{- 5 x + 4}}{2 (\frac{3 (2 (- 5 x + 4) - 5 (- 2 x - 1))}{2 (- 5 x + 4)})}$

Apply the distributive property.

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{- \frac{39 x}{- 5 x + 4}}{2 (\frac{3 (2 (- 5 x) + 2 \cdot 4 - 5 (- 2 x - 1))}{2 (- 5 x + 4)})}$

Multiply $- 5$ by $2$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{- \frac{39 x}{- 5 x + 4}}{2 (\frac{3 (- 10 x + 2 \cdot 4 - 5 (- 2 x - 1))}{2 (- 5 x + 4)})}$

Multiply $2$ by $4$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{- \frac{39 x}{- 5 x + 4}}{2 (\frac{3 (- 10 x + 8 - 5 (- 2 x - 1))}{2 (- 5 x + 4)})}$

Apply the distributive property.

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{- \frac{39 x}{- 5 x + 4}}{2 (\frac{3 (- 10 x + 8 - 5 (- 2 x) - 5 \cdot - 1)}{2 (- 5 x + 4)})}$

Multiply $- 2$ by $- 5$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{- \frac{39 x}{- 5 x + 4}}{2 (\frac{3 (- 10 x + 8 + 10 x - 5 \cdot - 1)}{2 (- 5 x + 4)})}$

Multiply $- 5$ by $- 1$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{- \frac{39 x}{- 5 x + 4}}{2 (\frac{3 (- 10 x + 8 + 10 x + 5)}{2 (- 5 x + 4)})}$

Add $- 10 x$ and $10 x$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{- \frac{39 x}{- 5 x + 4}}{2 (\frac{3 (0 + 8 + 5)}{2 (- 5 x + 4)})}$

Add $0$ and $8$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{- \frac{39 x}{- 5 x + 4}}{2 (\frac{3 (8 + 5)}{2 (- 5 x + 4)})}$

Add $8$ and $5$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{- \frac{39 x}{- 5 x + 4}}{2 (\frac{3 \cdot 13}{2 (- 5 x + 4)})}$

Multiply $3$ by $13$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{- \frac{39 x}{- 5 x + 4}}{2 (\frac{39}{2 (- 5 x + 4)})}$

Simplify terms.

Combine $2$ and $\frac{39}{2 (- 5 x + 4)}$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{- \frac{39 x}{- 5 x + 4}}{\frac{2 \cdot 39}{2 (- 5 x + 4)}}$

Multiply $2$ by $39$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{- \frac{39 x}{- 5 x + 4}}{\frac{78}{2 (- 5 x + 4)}}$

Reduce the expression $\frac{78}{2 (- 5 x + 4)}$ by cancelling the common factors.

Factor $2$ out of $78$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{- \frac{39 x}{- 5 x + 4}}{\frac{2 \cdot 39}{2 (- 5 x + 4)}}$

Cancel the common factor.

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{- \frac{39 x}{- 5 x + 4}}{\frac{2 \cdot 39}{2 (- 5 x + 4)}}$

Rewrite the expression.

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{- \frac{39 x}{- 5 x + 4}}{\frac{39}{- 5 x + 4}}$

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - (- \frac{39 x}{- 5 x + 4} \cdot \frac{- 5 x + 4}{39})$

Cancel the common factor of $39$ .

Move the leading negative in $- \frac{39 x}{- 5 x + 4}$ into the numerator.

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - (\frac{- 39 x}{- 5 x + 4} \cdot \frac{- 5 x + 4}{39})$

Factor $39$ out of $- 39 x$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - (\frac{39 (- x)}{- 5 x + 4} \cdot \frac{- 5 x + 4}{39})$

Cancel the common factor.

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - (\frac{39 (- x)}{- 5 x + 4} \cdot \frac{- 5 x + 4}{39})$

Rewrite the expression.

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - (\frac{- x}{- 5 x + 4} \cdot (- 5 x + 4))$

Move the negative in front of the fraction.

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - (- \frac{x}{- 5 x + 4} \cdot (- 5 x + 4))$

Apply the distributive property.

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - (- \frac{x}{- 5 x + 4} \cdot (- 5 x) - \frac{x}{- 5 x + 4} \cdot 4)$

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

Multiply $- 5$ by $- 1$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - (5 (\frac{x}{- 5 x + 4}) \cdot x - \frac{x}{- 5 x + 4} \cdot 4)$

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

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - (\frac{5 x}{- 5 x + 4} \cdot x - \frac{x}{- 5 x + 4} \cdot 4)$

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

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - (\frac{5 x \cdot x}{- 5 x + 4} - \frac{x}{- 5 x + 4} \cdot 4)$

Raise $x$ to the power of $1$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - (\frac{5 (x \cdot x)}{- 5 x + 4} - \frac{x}{- 5 x + 4} \cdot 4)$

Raise $x$ to the power of $1$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - (\frac{5 (x \cdot x)}{- 5 x + 4} - \frac{x}{- 5 x + 4} \cdot 4)$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - (\frac{5 x^{1 + 1}}{- 5 x + 4} - \frac{x}{- 5 x + 4} \cdot 4)$

Add $1$ and $1$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - (\frac{5 x^{2}}{- 5 x + 4} - \frac{x}{- 5 x + 4} \cdot 4)$

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

Multiply $4$ by $- 1$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - (\frac{5 x^{2}}{- 5 x + 4} - 4 \frac{x}{- 5 x + 4})$

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

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - (\frac{5 x^{2}}{- 5 x + 4} + \frac{- 4 x}{- 5 x + 4})$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{5 x^{2} - 4 x}{- 5 x + 4}$

Factor $x$ out of $5 x^{2} - 4 x$ .

Factor $x$ out of $5 x^{2}$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{x (5 x) - 4 x}{- 5 x + 4}$

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

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{x (5 x) + x \cdot - 4}{- 5 x + 4}$

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

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{x (5 x - 4)}{- 5 x + 4}$

Cancel the common factor of $5 x - 4$ and $- 5 x + 4$ .

Factor $- 1$ out of $5 x$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{x (- (- 5 x) - 4)}{- 5 x + 4}$

Rewrite $- 4$ as $- 1 (4)$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{x (- (- 5 x) - 1 \cdot 4)}{- 5 x + 4}$

Factor $- 1$ out of $- (- 5 x) - 1 (4)$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{x (- (- 5 x + 4))}{- 5 x + 4}$

Rewrite $- (- 5 x + 4)$ as $- 1 (- 5 x + 4)$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{x (- 1 (- 5 x + 4))}{- 5 x + 4}$

Cancel the common factor.

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - \frac{x (- 1 (- 5 x + 4))}{- 5 x + 4}$

Divide $x \cdot (- 1)$ by $1$ .

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - (x \cdot (- 1))$

Simplify the expression.

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

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = - (- 1 x)$

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

$f^{- 1} (\frac{- 6 x - 3}{- 10 x + 8}) = x$

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

Set up the composite result function.

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

Evaluate $f (- \frac{8 x + 3}{2 (3 - 5 x)})$ by substituting in the value of $f^{- 1}$ into $f$ .

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

Simplify the numerator.

Factor $- 3$ out of $- 6 (- \frac{8 x + 3}{2 (3 - 5 x)}) - 3$ .

Reorder $3$ and $- 5 x$ .

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{- 6 \cdot (- 1 \frac{8 x + 3}{2 (- 5 x + 3)}) - 3}{- 10 (- \frac{8 x + 3}{2 (3 - 5 x)}) + 8}$

Factor $- 3$ out of $- 6 \cdot - 1 \frac{8 x + 3}{2 (- 5 x + 3)}$ .

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{- 3 (2 \cdot (- 1 \frac{8 x + 3}{2 (- 5 x + 3)})) - 3}{- 10 (- \frac{8 x + 3}{2 (3 - 5 x)}) + 8}$

Factor $- 3$ out of $- 3$ .

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{- 3 (2 \cdot (- 1 \frac{8 x + 3}{2 (- 5 x + 3)})) - 3 \cdot 1}{- 10 (- \frac{8 x + 3}{2 (3 - 5 x)}) + 8}$

Factor $- 3$ out of $- 3 (2 \cdot - 1 \frac{8 x + 3}{2 (- 5 x + 3)}) - 3 (1)$ .

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{- 3 (2 \cdot (- 1 \frac{8 x + 3}{2 (- 5 x + 3)}) + 1)}{- 10 (- \frac{8 x + 3}{2 (3 - 5 x)}) + 8}$

Cancel the common factor of $2$ .

Factor $2$ out of $2 \cdot - 1$ .

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{- 3 (2 (- 1) (\frac{8 x + 3}{2 (- 5 x + 3)}) + 1)}{- 10 (- \frac{8 x + 3}{2 (3 - 5 x)}) + 8}$

Cancel the common factor.

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{- 3 (2 \cdot (- 1 \frac{8 x + 3}{2 (- 5 x + 3)}) + 1)}{- 10 (- \frac{8 x + 3}{2 (3 - 5 x)}) + 8}$

Rewrite the expression.

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{- 3 (- \frac{8 x + 3}{- 5 x + 3} + 1)}{- 10 (- \frac{8 x + 3}{2 (3 - 5 x)}) + 8}$

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Reorder terms.

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

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

Apply the distributive property.

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{- 3 \frac{- 5 x + 3 - (8 x) - 1 \cdot 3}{- 5 x + 3}}{- 10 (- \frac{8 x + 3}{2 (3 - 5 x)}) + 8}$

Multiply $8$ by $- 1$ .

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{- 3 \frac{- 5 x + 3 - 8 x - 1 \cdot 3}{- 5 x + 3}}{- 10 (- \frac{8 x + 3}{2 (3 - 5 x)}) + 8}$

Multiply $- 1$ by $3$ .

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

Subtract $8 x$ from $- 5 x$ .

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{- 3 \frac{- 13 x + 3 - 3}{- 5 x + 3}}{- 10 (- \frac{8 x + 3}{2 (3 - 5 x)}) + 8}$

Subtract $3$ from $3$ .

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{- 3 \frac{- 13 x + 0}{- 5 x + 3}}{- 10 (- \frac{8 x + 3}{2 (3 - 5 x)}) + 8}$

Add $- 13 x$ and $0$ .

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{- 3 \frac{- 13 x}{- 5 x + 3}}{- 10 (- \frac{8 x + 3}{2 (3 - 5 x)}) + 8}$

Move the negative in front of the fraction.

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{- 3 (- \frac{(13) x}{- 5 x + 3})}{- 10 (- \frac{8 x + 3}{2 (3 - 5 x)}) + 8}$

Remove unnecessary parentheses.

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{- 3 \cdot (- 1 \frac{13 x}{- 5 x + 3})}{- 10 (- \frac{8 x + 3}{2 (3 - 5 x)}) + 8}$

Combine exponents.

Factor out negative.

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{- (- 3 \frac{13 x}{- 5 x + 3})}{- 10 (- \frac{8 x + 3}{2 (3 - 5 x)}) + 8}$

Combine $- 3$ and $\frac{13 x}{- 5 x + 3}$ .

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{- \frac{- 3 (13 x)}{- 5 x + 3}}{- 10 (- \frac{8 x + 3}{2 (3 - 5 x)}) + 8}$

Multiply $13$ by $- 3$ .

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{- \frac{- 39 x}{- 5 x + 3}}{- 10 (- \frac{8 x + 3}{2 (3 - 5 x)}) + 8}$

Move the negative in front of the fraction.

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{\frac{39 x}{- 5 x + 3}}{- 10 (- \frac{8 x + 3}{2 (3 - 5 x)}) + 8}$

Simplify the denominator.

Factor $2$ out of $- 10 (- \frac{8 x + 3}{2 (3 - 5 x)}) + 8$ .

Factor $2$ out of $- 10 (- \frac{8 x + 3}{2 (3 - 5 x)})$ .

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{\frac{39 x}{- 5 x + 3}}{2 (- 5 (- \frac{8 x + 3}{2 (3 - 5 x)})) + 8}$

Factor $2$ out of $8$ .

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{\frac{39 x}{- 5 x + 3}}{2 (- 5 (- \frac{8 x + 3}{2 (3 - 5 x)})) + 2 (4)}$

Factor $2$ out of $2 (- 5 (- \frac{8 x + 3}{2 (3 - 5 x)})) + 2 (4)$ .

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{\frac{39 x}{- 5 x + 3}}{2 (- 5 (- \frac{8 x + 3}{2 (3 - 5 x)}) + 4)}$

Multiply $- 5 (- \frac{8 x + 3}{2 (3 - 5 x)})$ .

Multiply $- 1$ by $- 5$ .

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{\frac{39 x}{- 5 x + 3}}{2 (5 (\frac{8 x + 3}{2 (3 - 5 x)}) + 4)}$

Combine $5$ and $\frac{8 x + 3}{2 (3 - 5 x)}$ .

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{\frac{39 x}{- 5 x + 3}}{2 (\frac{5 (8 x + 3)}{2 (3 - 5 x)} + 4)}$

To write $4$ as a fraction with a common denominator, multiply by $\frac{2 (3 - 5 x)}{2 (3 - 5 x)}$ .

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{\frac{39 x}{- 5 x + 3}}{2 (\frac{5 (8 x + 3)}{2 (3 - 5 x)} + 4 \cdot \frac{2 (3 - 5 x)}{2 (3 - 5 x)})}$

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

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{\frac{39 x}{- 5 x + 3}}{2 (\frac{5 (8 x + 3)}{2 (3 - 5 x)} + \frac{4 (2 (3 - 5 x))}{2 (3 - 5 x)})}$

Combine the numerators over the common denominator.

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{\frac{39 x}{- 5 x + 3}}{2 (\frac{5 (8 x + 3) + 4 (2 (3 - 5 x))}{2 (3 - 5 x)})}$

Rewrite $\frac{5 (8 x + 3) + 4 (2 (3 - 5 x))}{2 (3 - 5 x)}$ in a factored form.

Apply the distributive property.

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{\frac{39 x}{- 5 x + 3}}{2 (\frac{5 (8 x) + 5 \cdot 3 + 4 (2 (3 - 5 x))}{2 (3 - 5 x)})}$

Multiply $8$ by $5$ .

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{\frac{39 x}{- 5 x + 3}}{2 (\frac{40 x + 5 \cdot 3 + 4 (2 (3 - 5 x))}{2 (3 - 5 x)})}$

Multiply $5$ by $3$ .

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{\frac{39 x}{- 5 x + 3}}{2 (\frac{40 x + 15 + 4 (2 (3 - 5 x))}{2 (3 - 5 x)})}$

Apply the distributive property.

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{\frac{39 x}{- 5 x + 3}}{2 (\frac{40 x + 15 + 4 (2 \cdot 3 + 2 (- 5 x))}{2 (3 - 5 x)})}$

Multiply $2$ by $3$ .

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{\frac{39 x}{- 5 x + 3}}{2 (\frac{40 x + 15 + 4 (6 + 2 (- 5 x))}{2 (3 - 5 x)})}$

Multiply $- 5$ by $2$ .

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{\frac{39 x}{- 5 x + 3}}{2 (\frac{40 x + 15 + 4 (6 - 10 x)}{2 (3 - 5 x)})}$

Apply the distributive property.

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{\frac{39 x}{- 5 x + 3}}{2 (\frac{40 x + 15 + 4 \cdot 6 + 4 (- 10 x)}{2 (3 - 5 x)})}$

Multiply $4$ by $6$ .

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{\frac{39 x}{- 5 x + 3}}{2 (\frac{40 x + 15 + 24 + 4 (- 10 x)}{2 (3 - 5 x)})}$

Multiply $- 10$ by $4$ .

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{\frac{39 x}{- 5 x + 3}}{2 (\frac{40 x + 15 + 24 - 40 x}{2 (3 - 5 x)})}$

Subtract $40 x$ from $40 x$ .

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{\frac{39 x}{- 5 x + 3}}{2 (\frac{0 + 15 + 24}{2 (3 - 5 x)})}$

Add $0$ and $15$ .

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{\frac{39 x}{- 5 x + 3}}{2 (\frac{15 + 24}{2 (3 - 5 x)})}$

Add $15$ and $24$ .

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{\frac{39 x}{- 5 x + 3}}{2 (\frac{39}{2 (3 - 5 x)})}$

Simplify terms.

Combine $2$ and $\frac{39}{2 (3 - 5 x)}$ .

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{\frac{39 x}{- 5 x + 3}}{\frac{2 \cdot 39}{2 (3 - 5 x)}}$

Multiply $2$ by $39$ .

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{\frac{39 x}{- 5 x + 3}}{\frac{78}{2 (3 - 5 x)}}$

Reduce the expression $\frac{78}{2 (3 - 5 x)}$ by cancelling the common factors.

Factor $2$ out of $78$ .

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{\frac{39 x}{- 5 x + 3}}{\frac{2 \cdot 39}{2 (3 - 5 x)}}$

Cancel the common factor.

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{\frac{39 x}{- 5 x + 3}}{\frac{2 \cdot 39}{2 (3 - 5 x)}}$

Rewrite the expression.

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{\frac{39 x}{- 5 x + 3}}{\frac{39}{3 - 5 x}}$

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{39 x}{- 5 x + 3} \cdot \frac{3 - 5 x}{39}$

Cancel the common factor of $39$ .

Factor $39$ out of $39 x$ .

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{39 (x)}{- 5 x + 3} \cdot \frac{3 - 5 x}{39}$

Cancel the common factor.

$f (- \frac{8 x + 3}{2 (3 - 5 x)}) = \frac{39 x}{- 5 x + 3} \cdot \frac{3 - 5 x}{39}$

Rewrite the expression.

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

Multiply $\frac{x}{- 5 x + 3}$ by $3 - 5 x$ .

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

Cancel the common factor of $3 - 5 x$ and $- 5 x + 3$ .

Reorder terms.

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

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{8 x + 3}{2 (3 - 5 x)}$ is the inverse of $f (x) = \frac{- 6 x - 3}{- 10 x + 8}$ .

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

Find the Inverse (-6x-3)/(-10x+8)

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