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

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

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

Multiply the equation by $5 y - 6$ .

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

Simplify the left side.

Apply the distributive property.

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

Simplify the right side.

Cancel the common factor of $5 y - 6$ .

Move the leading negative in $- \frac{8 y}{5 y - 6}$ into the numerator.

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

Cancel the common factor.

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

Rewrite the expression.

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

Solve for $y$ .

Add $8 y$ to both sides of the equation.

$5 y x - 6 x + 8 y = 0$

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

$5 y x + 8 y = 6 x$

Factor $y$ out of $5 y x + 8 y$ .

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

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

Factor $y$ out of $8 y$ .

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

Factor $y$ out of $y (5 x) + y \cdot 8$ .

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

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

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

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

Simplify the left side.

Cancel the common factor of $5 x + 8$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Step 3

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

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

Step 4

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

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

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

Simplify the numerator.

Multiply $- 1$ by $6$ .

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

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

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

Simplify the denominator.

Multiply $5 (- \frac{8 x}{5 x - 6})$ .

Multiply $- 1$ by $5$ .

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

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

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

Multiply $8$ by $- 5$ .

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

Move the negative in front of the fraction.

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

To write $8$ as a fraction with a common denominator, multiply by $\frac{5 x - 6}{5 x - 6}$ .

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

Combine the numerators over the common denominator.

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

Rewrite $\frac{- 40 x + 8 (5 x - 6)}{5 x - 6}$ in a factored form.

Factor $8$ out of $- 40 x + 8 (5 x - 6)$ .

Factor $8$ out of $- 40 x$ .

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

Factor $8$ out of $8 (- 5 x) + 8 (5 x - 6)$ .

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

Add $- 5 x$ and $5 x$ .

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

Subtract $6$ from $0$ .

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

Multiply $8$ by $- 6$ .

$f^{- 1} (- \frac{8 x}{5 x - 6}) = \frac{\frac{- 6 (8 x)}{5 x - 6}}{\frac{- 48}{5 x - 6}}$

Move the negative in front of the fraction.

$f^{- 1} (- \frac{8 x}{5 x - 6}) = \frac{\frac{- 6 (8 x)}{5 x - 6}}{- \frac{48}{5 x - 6}}$

Multiply $- 6$ by $8$ .

$f^{- 1} (- \frac{8 x}{5 x - 6}) = \frac{\frac{- 48 x}{5 x - 6}}{- \frac{48}{5 x - 6}}$

Move the negative in front of the fraction.

$f^{- 1} (- \frac{8 x}{5 x - 6}) = \frac{- \frac{48 x}{5 x - 6}}{- \frac{48}{5 x - 6}}$

Dividing two negative values results in a positive value.

$f^{- 1} (- \frac{8 x}{5 x - 6}) = \frac{\frac{48 x}{5 x - 6}}{\frac{48}{5 x - 6}}$

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (- \frac{8 x}{5 x - 6}) = \frac{48 x}{5 x - 6} \cdot \frac{5 x - 6}{48}$

Cancel the common factor of $48$ .

Factor $48$ out of $48 x$ .

$f^{- 1} (- \frac{8 x}{5 x - 6}) = \frac{48 (x)}{5 x - 6} \cdot \frac{5 x - 6}{48}$

Cancel the common factor.

$f^{- 1} (- \frac{8 x}{5 x - 6}) = \frac{48 x}{5 x - 6} \cdot \frac{5 x - 6}{48}$

Rewrite the expression.

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

Cancel the common factor of $5 x - 6$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

Set up the composite result function.

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

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

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

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

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

Simplify the denominator.

Multiply $5 \frac{6 x}{5 x + 8}$ .

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

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

Multiply $6$ by $5$ .

$f (\frac{6 x}{5 x + 8}) = - \frac{\frac{8 (6 x)}{5 x + 8}}{\frac{30 x}{5 x + 8} - 6}$

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{5 x + 8}{5 x + 8}$ .

$f (\frac{6 x}{5 x + 8}) = - \frac{\frac{8 (6 x)}{5 x + 8}}{\frac{30 x}{5 x + 8} - 6 \cdot \frac{5 x + 8}{5 x + 8}}$

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

$f (\frac{6 x}{5 x + 8}) = - \frac{\frac{8 (6 x)}{5 x + 8}}{\frac{30 x}{5 x + 8} + \frac{- 6 (5 x + 8)}{5 x + 8}}$

Combine the numerators over the common denominator.

$f (\frac{6 x}{5 x + 8}) = - \frac{\frac{8 (6 x)}{5 x + 8}}{\frac{30 x - 6 (5 x + 8)}{5 x + 8}}$

Rewrite $\frac{30 x - 6 (5 x + 8)}{5 x + 8}$ in a factored form.

Factor $6$ out of $30 x - 6 (5 x + 8)$ .

Factor $6$ out of $30 x$ .

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

Factor $6$ out of $- 6 (5 x + 8)$ .

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

Factor $6$ out of $6 (5 x) + 6 (- (5 x + 8))$ .

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

Apply the distributive property.

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

Multiply $5$ by $- 1$ .

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

Multiply $- 1$ by $8$ .

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

Subtract $5 x$ from $5 x$ .

$f (\frac{6 x}{5 x + 8}) = - \frac{\frac{8 (6 x)}{5 x + 8}}{\frac{6 (0 - 8)}{5 x + 8}}$

Subtract $8$ from $0$ .

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

Multiply $6$ by $- 8$ .

$f (\frac{6 x}{5 x + 8}) = - \frac{\frac{8 (6 x)}{5 x + 8}}{\frac{- 48}{5 x + 8}}$

Move the negative in front of the fraction.

$f (\frac{6 x}{5 x + 8}) = - \frac{\frac{8 (6 x)}{5 x + 8}}{- \frac{48}{5 x + 8}}$

Multiply $8$ by $6$ .

$f (\frac{6 x}{5 x + 8}) = - \frac{\frac{48 x}{5 x + 8}}{- \frac{48}{5 x + 8}}$

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{6 x}{5 x + 8}) = - (\frac{48 x}{5 x + 8} \cdot (- \frac{5 x + 8}{48}))$

Rewrite using the commutative property of multiplication.

$f (\frac{6 x}{5 x + 8}) = - (- \frac{48 x}{5 x + 8} \cdot \frac{5 x + 8}{48})$

Cancel the common factor of $48$ .

Move the leading negative in $- \frac{48 x}{5 x + 8}$ into the numerator.

$f (\frac{6 x}{5 x + 8}) = - (\frac{- 48 x}{5 x + 8} \cdot \frac{5 x + 8}{48})$

Factor $48$ out of $- 48 x$ .

$f (\frac{6 x}{5 x + 8}) = - (\frac{48 (- x)}{5 x + 8} \cdot \frac{5 x + 8}{48})$

Cancel the common factor.

$f (\frac{6 x}{5 x + 8}) = - (\frac{48 (- x)}{5 x + 8} \cdot \frac{5 x + 8}{48})$

Rewrite the expression.

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

Cancel the common factor of $5 x + 8$ .

Cancel the common factor.

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

Rewrite the expression.

$f (\frac{6 x}{5 x + 8}) = x$

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

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

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

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