Find the Inverse (8x+9)/(5x-8)

$\frac{8 x + 9}{5 x - 8}$

Step 1

Interchange the variables.

$x = \frac{8 y + 9}{5 y - 8}$

Step 2

Solve for $y$ .

Multiply the equation by $5 y - 8$ .

$(5 y - 8) (x) = (\frac{8 y + 9}{5 y - 8}) (5 y - 8)$

Simplify the left side.

Apply the distributive property.

$5 y x - 8 x = (\frac{8 y + 9}{5 y - 8}) (5 y - 8)$

Simplify the right side.

Cancel the common factor of $5 y - 8$ .

Cancel the common factor.

$5 y x - 8 x = \frac{8 y + 9}{5 y - 8} (5 y - 8)$

Rewrite the expression.

$5 y x - 8 x = 8 y + 9$

Solve for $y$ .

Subtract $8 y$ from both sides of the equation.

$5 y x - 8 x - 8 y = 9$

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

$5 y x - 8 y = 9 + 8 x$

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

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

$y (5 x) - 8 y = 9 + 8 x$

Factor $y$ out of $- 8 y$ .

$y (5 x) + y \cdot - 8 = 9 + 8 x$

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

$y (5 x - 8) = 9 + 8 x$

Divide each term in $y (5 x - 8) = 9 + 8 x$ by $5 x - 8$ and simplify.

Divide each term in $y (5 x - 8) = 9 + 8 x$ by $5 x - 8$ .

$\frac{y (5 x - 8)}{5 x - 8} = \frac{9}{5 x - 8} + \frac{8 x}{5 x - 8}$

Simplify the left side.

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

Cancel the common factor.

$\frac{y (5 x - 8)}{5 x - 8} = \frac{9}{5 x - 8} + \frac{8 x}{5 x - 8}$

Divide $y$ by $1$ .

$y = \frac{9}{5 x - 8} + \frac{8 x}{5 x - 8}$

Simplify the right side.

Combine the numerators over the common denominator.

$y = \frac{9 + 8 x}{5 x - 8}$

Step 3

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

$f^{- 1} (x) = \frac{9 + 8 x}{5 x - 8}$

Step 4

Verify if $f^{- 1} (x) = \frac{9 + 8 x}{5 x - 8}$ is the inverse of $f (x) = \frac{8 x + 9}{5 x - 8}$ .

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 $f^{- 1} (\frac{8 x + 9}{5 x - 8})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{8 x + 9}{5 x - 8}) = \frac{9 + 8 (\frac{8 x + 9}{5 x - 8})}{5 (\frac{8 x + 9}{5 x - 8}) - 8}$

Simplify the numerator.

Combine $8$ and $\frac{8 x + 9}{5 x - 8}$ .

$f^{- 1} (\frac{8 x + 9}{5 x - 8}) = \frac{9 + \frac{8 (8 x + 9)}{5 x - 8}}{5 (\frac{8 x + 9}{5 x - 8}) - 8}$

To write $9$ as a fraction with a common denominator, multiply by $\frac{5 x - 8}{5 x - 8}$ .

$f^{- 1} (\frac{8 x + 9}{5 x - 8}) = \frac{\frac{9 (5 x - 8)}{5 x - 8} + \frac{8 (8 x + 9)}{5 x - 8}}{5 (\frac{8 x + 9}{5 x - 8}) - 8}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{8 x + 9}{5 x - 8}) = \frac{\frac{9 (5 x - 8) + 8 (8 x + 9)}{5 x - 8}}{5 (\frac{8 x + 9}{5 x - 8}) - 8}$

Rewrite $\frac{9 (5 x - 8) + 8 (8 x + 9)}{5 x - 8}$ in a factored form.

Apply the distributive property.

$f^{- 1} (\frac{8 x + 9}{5 x - 8}) = \frac{\frac{9 (5 x) + 9 \cdot - 8 + 8 (8 x + 9)}{5 x - 8}}{5 (\frac{8 x + 9}{5 x - 8}) - 8}$

Multiply $5$ by $9$ .

$f^{- 1} (\frac{8 x + 9}{5 x - 8}) = \frac{\frac{45 x + 9 \cdot - 8 + 8 (8 x + 9)}{5 x - 8}}{5 (\frac{8 x + 9}{5 x - 8}) - 8}$

Multiply $9$ by $- 8$ .

$f^{- 1} (\frac{8 x + 9}{5 x - 8}) = \frac{\frac{45 x - 72 + 8 (8 x + 9)}{5 x - 8}}{5 (\frac{8 x + 9}{5 x - 8}) - 8}$

Apply the distributive property.

$f^{- 1} (\frac{8 x + 9}{5 x - 8}) = \frac{\frac{45 x - 72 + 8 (8 x) + 8 \cdot 9}{5 x - 8}}{5 (\frac{8 x + 9}{5 x - 8}) - 8}$

Multiply $8$ by $8$ .

$f^{- 1} (\frac{8 x + 9}{5 x - 8}) = \frac{\frac{45 x - 72 + 64 x + 8 \cdot 9}{5 x - 8}}{5 (\frac{8 x + 9}{5 x - 8}) - 8}$

Multiply $8$ by $9$ .

$f^{- 1} (\frac{8 x + 9}{5 x - 8}) = \frac{\frac{45 x - 72 + 64 x + 72}{5 x - 8}}{5 (\frac{8 x + 9}{5 x - 8}) - 8}$

Add $45 x$ and $64 x$ .

$f^{- 1} (\frac{8 x + 9}{5 x - 8}) = \frac{\frac{109 x - 72 + 72}{5 x - 8}}{5 (\frac{8 x + 9}{5 x - 8}) - 8}$

Add $- 72$ and $72$ .

$f^{- 1} (\frac{8 x + 9}{5 x - 8}) = \frac{\frac{109 x + 0}{5 x - 8}}{5 (\frac{8 x + 9}{5 x - 8}) - 8}$

Add $109 x$ and $0$ .

$f^{- 1} (\frac{8 x + 9}{5 x - 8}) = \frac{\frac{109 x}{5 x - 8}}{5 (\frac{8 x + 9}{5 x - 8}) - 8}$

Simplify the denominator.

Combine $5$ and $\frac{8 x + 9}{5 x - 8}$ .

$f^{- 1} (\frac{8 x + 9}{5 x - 8}) = \frac{\frac{109 x}{5 x - 8}}{\frac{5 (8 x + 9)}{5 x - 8} - 8}$

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

$f^{- 1} (\frac{8 x + 9}{5 x - 8}) = \frac{\frac{109 x}{5 x - 8}}{\frac{5 (8 x + 9)}{5 x - 8} - 8 \cdot \frac{5 x - 8}{5 x - 8}}$

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

$f^{- 1} (\frac{8 x + 9}{5 x - 8}) = \frac{\frac{109 x}{5 x - 8}}{\frac{5 (8 x + 9)}{5 x - 8} + \frac{- 8 (5 x - 8)}{5 x - 8}}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{8 x + 9}{5 x - 8}) = \frac{\frac{109 x}{5 x - 8}}{\frac{5 (8 x + 9) - 8 (5 x - 8)}{5 x - 8}}$

Rewrite $\frac{5 (8 x + 9) - 8 (5 x - 8)}{5 x - 8}$ in a factored form.

Apply the distributive property.

$f^{- 1} (\frac{8 x + 9}{5 x - 8}) = \frac{\frac{109 x}{5 x - 8}}{\frac{5 (8 x) + 5 \cdot 9 - 8 (5 x - 8)}{5 x - 8}}$

Multiply $8$ by $5$ .

$f^{- 1} (\frac{8 x + 9}{5 x - 8}) = \frac{\frac{109 x}{5 x - 8}}{\frac{40 x + 5 \cdot 9 - 8 (5 x - 8)}{5 x - 8}}$

Multiply $5$ by $9$ .

$f^{- 1} (\frac{8 x + 9}{5 x - 8}) = \frac{\frac{109 x}{5 x - 8}}{\frac{40 x + 45 - 8 (5 x - 8)}{5 x - 8}}$

Apply the distributive property.

$f^{- 1} (\frac{8 x + 9}{5 x - 8}) = \frac{\frac{109 x}{5 x - 8}}{\frac{40 x + 45 - 8 (5 x) - 8 \cdot - 8}{5 x - 8}}$

Multiply $5$ by $- 8$ .

$f^{- 1} (\frac{8 x + 9}{5 x - 8}) = \frac{\frac{109 x}{5 x - 8}}{\frac{40 x + 45 - 40 x - 8 \cdot - 8}{5 x - 8}}$

Multiply $- 8$ by $- 8$ .

$f^{- 1} (\frac{8 x + 9}{5 x - 8}) = \frac{\frac{109 x}{5 x - 8}}{\frac{40 x + 45 - 40 x + 64}{5 x - 8}}$

Subtract $40 x$ from $40 x$ .

$f^{- 1} (\frac{8 x + 9}{5 x - 8}) = \frac{\frac{109 x}{5 x - 8}}{\frac{0 + 45 + 64}{5 x - 8}}$

Add $0$ and $45$ .

$f^{- 1} (\frac{8 x + 9}{5 x - 8}) = \frac{\frac{109 x}{5 x - 8}}{\frac{45 + 64}{5 x - 8}}$

Add $45$ and $64$ .

$f^{- 1} (\frac{8 x + 9}{5 x - 8}) = \frac{\frac{109 x}{5 x - 8}}{\frac{109}{5 x - 8}}$

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{8 x + 9}{5 x - 8}) = \frac{109 x}{5 x - 8} \cdot \frac{5 x - 8}{109}$

Cancel the common factor of $109$ .

Factor $109$ out of $109 x$ .

$f^{- 1} (\frac{8 x + 9}{5 x - 8}) = \frac{109 (x)}{5 x - 8} \cdot \frac{5 x - 8}{109}$

Cancel the common factor.

$f^{- 1} (\frac{8 x + 9}{5 x - 8}) = \frac{109 x}{5 x - 8} \cdot \frac{5 x - 8}{109}$

Rewrite the expression.

$f^{- 1} (\frac{8 x + 9}{5 x - 8}) = \frac{x}{5 x - 8} \cdot (5 x - 8)$

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

Cancel the common factor.

$f^{- 1} (\frac{8 x + 9}{5 x - 8}) = \frac{x}{5 x - 8} \cdot (5 x - 8)$

Rewrite the expression.

$f^{- 1} (\frac{8 x + 9}{5 x - 8}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{9 + 8 x}{5 x - 8})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{9 + 8 x}{5 x - 8}) = \frac{8 (\frac{9 + 8 x}{5 x - 8}) + 9}{5 (\frac{9 + 8 x}{5 x - 8}) - 8}$

Simplify the numerator.

Combine $8$ and $\frac{9 + 8 x}{5 x - 8}$ .

$f (\frac{9 + 8 x}{5 x - 8}) = \frac{\frac{8 (9 + 8 x)}{5 x - 8} + 9}{5 (\frac{9 + 8 x}{5 x - 8}) - 8}$

To write $9$ as a fraction with a common denominator, multiply by $\frac{5 x - 8}{5 x - 8}$ .

$f (\frac{9 + 8 x}{5 x - 8}) = \frac{\frac{8 (9 + 8 x)}{5 x - 8} + \frac{9 (5 x - 8)}{5 x - 8}}{5 (\frac{9 + 8 x}{5 x - 8}) - 8}$

Combine the numerators over the common denominator.

$f (\frac{9 + 8 x}{5 x - 8}) = \frac{\frac{8 (9 + 8 x) + 9 (5 x - 8)}{5 x - 8}}{5 (\frac{9 + 8 x}{5 x - 8}) - 8}$

Reorder terms.

$f (\frac{9 + 8 x}{5 x - 8}) = \frac{\frac{9 (5 x - 8) + 8 (9 + 8 x)}{5 x - 8}}{5 (\frac{9 + 8 x}{5 x - 8}) - 8}$

Rewrite $\frac{9 (5 x - 8) + 8 (9 + 8 x)}{5 x - 8}$ in a factored form.

Apply the distributive property.

$f (\frac{9 + 8 x}{5 x - 8}) = \frac{\frac{9 (5 x) + 9 \cdot - 8 + 8 (9 + 8 x)}{5 x - 8}}{5 (\frac{9 + 8 x}{5 x - 8}) - 8}$

Multiply $5$ by $9$ .

$f (\frac{9 + 8 x}{5 x - 8}) = \frac{\frac{45 x + 9 \cdot - 8 + 8 (9 + 8 x)}{5 x - 8}}{5 (\frac{9 + 8 x}{5 x - 8}) - 8}$

Multiply $9$ by $- 8$ .

$f (\frac{9 + 8 x}{5 x - 8}) = \frac{\frac{45 x - 72 + 8 (9 + 8 x)}{5 x - 8}}{5 (\frac{9 + 8 x}{5 x - 8}) - 8}$

Apply the distributive property.

$f (\frac{9 + 8 x}{5 x - 8}) = \frac{\frac{45 x - 72 + 8 \cdot 9 + 8 (8 x)}{5 x - 8}}{5 (\frac{9 + 8 x}{5 x - 8}) - 8}$

Multiply $8$ by $9$ .

$f (\frac{9 + 8 x}{5 x - 8}) = \frac{\frac{45 x - 72 + 72 + 8 (8 x)}{5 x - 8}}{5 (\frac{9 + 8 x}{5 x - 8}) - 8}$

Multiply $8$ by $8$ .

$f (\frac{9 + 8 x}{5 x - 8}) = \frac{\frac{45 x - 72 + 72 + 64 x}{5 x - 8}}{5 (\frac{9 + 8 x}{5 x - 8}) - 8}$

Add $45 x$ and $64 x$ .

$f (\frac{9 + 8 x}{5 x - 8}) = \frac{\frac{109 x - 72 + 72}{5 x - 8}}{5 (\frac{9 + 8 x}{5 x - 8}) - 8}$

Add $- 72$ and $72$ .

$f (\frac{9 + 8 x}{5 x - 8}) = \frac{\frac{109 x + 0}{5 x - 8}}{5 (\frac{9 + 8 x}{5 x - 8}) - 8}$

Add $109 x$ and $0$ .

$f (\frac{9 + 8 x}{5 x - 8}) = \frac{\frac{109 x}{5 x - 8}}{5 (\frac{9 + 8 x}{5 x - 8}) - 8}$

Simplify the denominator.

Combine $5$ and $\frac{9 + 8 x}{5 x - 8}$ .

$f (\frac{9 + 8 x}{5 x - 8}) = \frac{\frac{109 x}{5 x - 8}}{\frac{5 (9 + 8 x)}{5 x - 8} - 8}$

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

$f (\frac{9 + 8 x}{5 x - 8}) = \frac{\frac{109 x}{5 x - 8}}{\frac{5 (9 + 8 x)}{5 x - 8} - 8 \cdot \frac{5 x - 8}{5 x - 8}}$

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

$f (\frac{9 + 8 x}{5 x - 8}) = \frac{\frac{109 x}{5 x - 8}}{\frac{5 (9 + 8 x)}{5 x - 8} + \frac{- 8 (5 x - 8)}{5 x - 8}}$

Combine the numerators over the common denominator.

$f (\frac{9 + 8 x}{5 x - 8}) = \frac{\frac{109 x}{5 x - 8}}{\frac{5 (9 + 8 x) - 8 (5 x - 8)}{5 x - 8}}$

Rewrite $\frac{5 (9 + 8 x) - 8 (5 x - 8)}{5 x - 8}$ in a factored form.

Apply the distributive property.

$f (\frac{9 + 8 x}{5 x - 8}) = \frac{\frac{109 x}{5 x - 8}}{\frac{5 \cdot 9 + 5 (8 x) - 8 (5 x - 8)}{5 x - 8}}$

Multiply $5$ by $9$ .

$f (\frac{9 + 8 x}{5 x - 8}) = \frac{\frac{109 x}{5 x - 8}}{\frac{45 + 5 (8 x) - 8 (5 x - 8)}{5 x - 8}}$

Multiply $8$ by $5$ .

$f (\frac{9 + 8 x}{5 x - 8}) = \frac{\frac{109 x}{5 x - 8}}{\frac{45 + 40 x - 8 (5 x - 8)}{5 x - 8}}$

Apply the distributive property.

$f (\frac{9 + 8 x}{5 x - 8}) = \frac{\frac{109 x}{5 x - 8}}{\frac{45 + 40 x - 8 (5 x) - 8 \cdot - 8}{5 x - 8}}$

Multiply $5$ by $- 8$ .

$f (\frac{9 + 8 x}{5 x - 8}) = \frac{\frac{109 x}{5 x - 8}}{\frac{45 + 40 x - 40 x - 8 \cdot - 8}{5 x - 8}}$

Multiply $- 8$ by $- 8$ .

$f (\frac{9 + 8 x}{5 x - 8}) = \frac{\frac{109 x}{5 x - 8}}{\frac{45 + 40 x - 40 x + 64}{5 x - 8}}$

Add $45$ and $64$ .

$f (\frac{9 + 8 x}{5 x - 8}) = \frac{\frac{109 x}{5 x - 8}}{\frac{40 x - 40 x + 109}{5 x - 8}}$

Subtract $40 x$ from $40 x$ .

$f (\frac{9 + 8 x}{5 x - 8}) = \frac{\frac{109 x}{5 x - 8}}{\frac{0 + 109}{5 x - 8}}$

Add $0$ and $109$ .

$f (\frac{9 + 8 x}{5 x - 8}) = \frac{\frac{109 x}{5 x - 8}}{\frac{109}{5 x - 8}}$

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{9 + 8 x}{5 x - 8}) = \frac{109 x}{5 x - 8} \cdot \frac{5 x - 8}{109}$

Cancel the common factor of $109$ .

Factor $109$ out of $109 x$ .

$f (\frac{9 + 8 x}{5 x - 8}) = \frac{109 (x)}{5 x - 8} \cdot \frac{5 x - 8}{109}$

Cancel the common factor.

$f (\frac{9 + 8 x}{5 x - 8}) = \frac{109 x}{5 x - 8} \cdot \frac{5 x - 8}{109}$

Rewrite the expression.

$f (\frac{9 + 8 x}{5 x - 8}) = \frac{x}{5 x - 8} \cdot (5 x - 8)$

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

Cancel the common factor.

$f (\frac{9 + 8 x}{5 x - 8}) = \frac{x}{5 x - 8} \cdot (5 x - 8)$

Rewrite the expression.

$f (\frac{9 + 8 x}{5 x - 8}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{9 + 8 x}{5 x - 8}$ is the inverse of $f (x) = \frac{8 x + 9}{5 x - 8}$ .

$f^{- 1} (x) = \frac{9 + 8 x}{5 x - 8}$

Find the Inverse (8x+9)/(5x-8)

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