Find the Inverse (6x-5)/(x+9)

$\frac{6 x - 5}{x + 9}$

Step 1

Interchange the variables.

$x = \frac{6 y - 5}{y + 9}$

Step 2

Solve for $y$ .

Rewrite the equation as $\frac{6 y - 5}{y + 9} = x$ .

$\frac{6 y - 5}{y + 9} = x$

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.

$y + 9, 1$

Remove parentheses.

$y + 9, 1$

The LCM of one and any expression is the expression.

$y + 9$

Multiply each term in $\frac{6 y - 5}{y + 9} = x$ by $y + 9$ to eliminate the fractions.

Multiply each term in $\frac{6 y - 5}{y + 9} = x$ by $y + 9$ .

$\frac{6 y - 5}{y + 9} (y + 9) = x (y + 9)$

Simplify the left side.

Cancel the common factor of $y + 9$ .

Cancel the common factor.

$\frac{6 y - 5}{y + 9} (y + 9) = x (y + 9)$

Rewrite the expression.

$6 y - 5 = x (y + 9)$

Simplify the right side.

Apply the distributive property.

$6 y - 5 = x y + x \cdot 9$

Move $9$ to the left of $x$ .

$6 y - 5 = x y + 9 x$

Solve the equation.

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

$6 y - 5 - x y = 9 x$

Add $5$ to both sides of the equation.

$6 y - x y = 9 x + 5$

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

Factor $y$ out of $6 y$ .

$y \cdot 6 - x y = 9 x + 5$

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

$y \cdot 6 + y (- x) = 9 x + 5$

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

$y (6 - x) = 9 x + 5$

Divide each term in $y (6 - x) = 9 x + 5$ by $6 - x$ and simplify.

Divide each term in $y (6 - x) = 9 x + 5$ by $6 - x$ .

$\frac{y (6 - x)}{6 - x} = \frac{9 x}{6 - x} + \frac{5}{6 - x}$

Simplify the left side.

Cancel the common factor of $6 - x$ .

Cancel the common factor.

$\frac{y (6 - x)}{6 - x} = \frac{9 x}{6 - x} + \frac{5}{6 - x}$

Divide $y$ by $1$ .

$y = \frac{9 x}{6 - x} + \frac{5}{6 - x}$

Simplify the right side.

Combine the numerators over the common denominator.

$y = \frac{9 x + 5}{6 - x}$

Step 3

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

$f^{- 1} (x) = \frac{9 x + 5}{6 - x}$

Step 4

Verify if $f^{- 1} (x) = \frac{9 x + 5}{6 - x}$ is the inverse of $f (x) = \frac{6 x - 5}{x + 9}$ .

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 $f^{- 1} (\frac{6 x - 5}{x + 9})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{6 x - 5}{x + 9}) = \frac{9 (\frac{6 x - 5}{x + 9}) + 5}{6 - (\frac{6 x - 5}{x + 9})}$

Multiply the numerator and denominator of the complex fraction by $x + 9$ .

Multiply $\frac{9 \frac{6 x - 5}{x + 9} + 5}{6 - \frac{6 x - 5}{x + 9}}$ by $\frac{x + 9}{x + 9}$ .

$f^{- 1} (\frac{6 x - 5}{x + 9}) = \frac{x + 9}{x + 9} \cdot \frac{9 (\frac{6 x - 5}{x + 9}) + 5}{6 - \frac{6 x - 5}{x + 9}}$

Combine.

$f^{- 1} (\frac{6 x - 5}{x + 9}) = \frac{(x + 9) (9 (\frac{6 x - 5}{x + 9}) + 5)}{(x + 9) (6 - \frac{6 x - 5}{x + 9})}$

Apply the distributive property.

$f^{- 1} (\frac{6 x - 5}{x + 9}) = \frac{(x + 9) (9 (\frac{6 x - 5}{x + 9})) + (x + 9) \cdot 5}{(x + 9) \cdot 6 + (x + 9) (- \frac{6 x - 5}{x + 9})}$

Cancel the common factor of $x + 9$ .

Move the leading negative in $- \frac{6 x - 5}{x + 9}$ into the numerator.

$f^{- 1} (\frac{6 x - 5}{x + 9}) = \frac{(x + 9) (9 (\frac{6 x - 5}{x + 9})) + (x + 9) \cdot 5}{(x + 9) \cdot 6 + (x + 9) (\frac{- (6 x - 5)}{x + 9})}$

Cancel the common factor.

$f^{- 1} (\frac{6 x - 5}{x + 9}) = \frac{(x + 9) (9 (\frac{6 x - 5}{x + 9})) + (x + 9) \cdot 5}{(x + 9) \cdot 6 + (x + 9) (\frac{- (6 x - 5)}{x + 9})}$

Rewrite the expression.

$f^{- 1} (\frac{6 x - 5}{x + 9}) = \frac{(x + 9) (9 (\frac{6 x - 5}{x + 9})) + (x + 9) \cdot 5}{(x + 9) \cdot 6 - (6 x - 5)}$

Simplify the numerator.

Factor $x + 9$ out of $(x + 9) \cdot 9 \frac{6 x - 5}{x + 9} + (x + 9) \cdot 5$ .

Factor $x + 9$ out of $(x + 9) \cdot 9 \frac{6 x - 5}{x + 9}$ .

$f^{- 1} (\frac{6 x - 5}{x + 9}) = \frac{(x + 9) (9 (\frac{6 x - 5}{x + 9})) + (x + 9) \cdot 5}{(x + 9) \cdot 6 - (6 x - 5)}$

Factor $x + 9$ out of $(x + 9) \cdot 5$ .

$f^{- 1} (\frac{6 x - 5}{x + 9}) = \frac{(x + 9) (9 (\frac{6 x - 5}{x + 9})) + (x + 9) (5)}{(x + 9) \cdot 6 - (6 x - 5)}$

Factor $x + 9$ out of $(x + 9) (9 \frac{6 x - 5}{x + 9}) + (x + 9) (5)$ .

$f^{- 1} (\frac{6 x - 5}{x + 9}) = \frac{(x + 9) (9 (\frac{6 x - 5}{x + 9}) + 5)}{(x + 9) \cdot 6 - (6 x - 5)}$

Combine $9$ and $\frac{6 x - 5}{x + 9}$ .

$f^{- 1} (\frac{6 x - 5}{x + 9}) = \frac{(x + 9) (\frac{9 (6 x - 5)}{x + 9} + 5)}{(x + 9) \cdot 6 - (6 x - 5)}$

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

$f^{- 1} (\frac{6 x - 5}{x + 9}) = \frac{(x + 9) (\frac{9 (6 x - 5)}{x + 9} + \frac{5 (x + 9)}{x + 9})}{(x + 9) \cdot 6 - (6 x - 5)}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{6 x - 5}{x + 9}) = \frac{(x + 9) (\frac{9 (6 x - 5) + 5 (x + 9)}{x + 9})}{(x + 9) \cdot 6 - (6 x - 5)}$

Rewrite $\frac{9 (6 x - 5) + 5 (x + 9)}{x + 9}$ in a factored form.

Apply the distributive property.

$f^{- 1} (\frac{6 x - 5}{x + 9}) = \frac{(x + 9) (\frac{9 (6 x) + 9 \cdot - 5 + 5 (x + 9)}{x + 9})}{(x + 9) \cdot 6 - (6 x - 5)}$

Multiply $6$ by $9$ .

$f^{- 1} (\frac{6 x - 5}{x + 9}) = \frac{(x + 9) (\frac{54 x + 9 \cdot - 5 + 5 (x + 9)}{x + 9})}{(x + 9) \cdot 6 - (6 x - 5)}$

Multiply $9$ by $- 5$ .

$f^{- 1} (\frac{6 x - 5}{x + 9}) = \frac{(x + 9) (\frac{54 x - 45 + 5 (x + 9)}{x + 9})}{(x + 9) \cdot 6 - (6 x - 5)}$

Apply the distributive property.

$f^{- 1} (\frac{6 x - 5}{x + 9}) = \frac{(x + 9) (\frac{54 x - 45 + 5 x + 5 \cdot 9}{x + 9})}{(x + 9) \cdot 6 - (6 x - 5)}$

Multiply $5$ by $9$ .

$f^{- 1} (\frac{6 x - 5}{x + 9}) = \frac{(x + 9) (\frac{54 x - 45 + 5 x + 45}{x + 9})}{(x + 9) \cdot 6 - (6 x - 5)}$

Add $54 x$ and $5 x$ .

$f^{- 1} (\frac{6 x - 5}{x + 9}) = \frac{(x + 9) (\frac{59 x - 45 + 45}{x + 9})}{(x + 9) \cdot 6 - (6 x - 5)}$

Add $- 45$ and $45$ .

$f^{- 1} (\frac{6 x - 5}{x + 9}) = \frac{(x + 9) (\frac{59 x + 0}{x + 9})}{(x + 9) \cdot 6 - (6 x - 5)}$

Add $59 x$ and $0$ .

$f^{- 1} (\frac{6 x - 5}{x + 9}) = \frac{(x + 9) (\frac{59 x}{x + 9})}{(x + 9) \cdot 6 - (6 x - 5)}$

Simplify the denominator.

Apply the distributive property.

$f^{- 1} (\frac{6 x - 5}{x + 9}) = \frac{(x + 9) (\frac{59 x}{x + 9})}{x \cdot 6 + 9 \cdot 6 - (6 x - 5)}$

Move $6$ to the left of $x$ .

$f^{- 1} (\frac{6 x - 5}{x + 9}) = \frac{(x + 9) (\frac{59 x}{x + 9})}{6 \cdot x + 9 \cdot 6 - (6 x - 5)}$

Multiply $9$ by $6$ .

$f^{- 1} (\frac{6 x - 5}{x + 9}) = \frac{(x + 9) (\frac{59 x}{x + 9})}{6 \cdot x + 54 - (6 x - 5)}$

Apply the distributive property.

$f^{- 1} (\frac{6 x - 5}{x + 9}) = \frac{(x + 9) (\frac{59 x}{x + 9})}{6 x + 54 - (6 x) + 5}$

Multiply $6$ by $- 1$ .

$f^{- 1} (\frac{6 x - 5}{x + 9}) = \frac{(x + 9) (\frac{59 x}{x + 9})}{6 x + 54 - 6 x + 5}$

Multiply $- 1$ by $- 5$ .

$f^{- 1} (\frac{6 x - 5}{x + 9}) = \frac{(x + 9) (\frac{59 x}{x + 9})}{6 x + 54 - 6 x + 5}$

Subtract $6 x$ from $6 x$ .

$f^{- 1} (\frac{6 x - 5}{x + 9}) = \frac{(x + 9) (\frac{59 x}{x + 9})}{0 + 54 + 5}$

Add $0$ and $54$ .

$f^{- 1} (\frac{6 x - 5}{x + 9}) = \frac{(x + 9) (\frac{59 x}{x + 9})}{54 + 5}$

Add $54$ and $5$ .

$f^{- 1} (\frac{6 x - 5}{x + 9}) = \frac{(x + 9) (\frac{59 x}{x + 9})}{59}$

Reduce the expression by cancelling the common factors.

Factor $\frac{59 x}{x + 9}$ out of $\frac{(x + 9) \frac{59 x}{x + 9}}{59}$ .

$f^{- 1} (\frac{6 x - 5}{x + 9}) = \frac{59 x}{x + 9} \cdot \frac{x + 9}{59}$

Cancel the common factor of $59$ .

Factor $59$ out of $59 x$ .

$f^{- 1} (\frac{6 x - 5}{x + 9}) = \frac{59 (x)}{x + 9} \cdot \frac{x + 9}{59}$

Cancel the common factor.

$f^{- 1} (\frac{6 x - 5}{x + 9}) = \frac{59 x}{x + 9} \cdot \frac{x + 9}{59}$

Rewrite the expression.

$f^{- 1} (\frac{6 x - 5}{x + 9}) = \frac{x}{x + 9} \cdot (x + 9)$

Cancel the common factor of $x + 9$ .

Cancel the common factor.

$f^{- 1} (\frac{6 x - 5}{x + 9}) = \frac{x}{x + 9} \cdot (x + 9)$

Rewrite the expression.

$f^{- 1} (\frac{6 x - 5}{x + 9}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{9 x + 5}{6 - x})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{9 x + 5}{6 - x}) = \frac{6 (\frac{9 x + 5}{6 - x}) - 5}{(\frac{9 x + 5}{6 - x}) + 9}$

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

Multiply $\frac{6 \frac{9 x + 5}{6 - x} - 5}{\frac{9 x + 5}{6 - x} + 9}$ by $\frac{6 - x}{6 - x}$ .

$f (\frac{9 x + 5}{6 - x}) = \frac{6 - x}{6 - x} \cdot \frac{6 (\frac{9 x + 5}{6 - x}) - 5}{\frac{9 x + 5}{6 - x} + 9}$

Combine.

$f (\frac{9 x + 5}{6 - x}) = \frac{(6 - x) (6 (\frac{9 x + 5}{6 - x}) - 5)}{(6 - x) (\frac{9 x + 5}{6 - x} + 9)}$

Apply the distributive property.

$f (\frac{9 x + 5}{6 - x}) = \frac{(6 - x) (6 (\frac{9 x + 5}{6 - x})) + (6 - x) \cdot - 5}{(6 - x) (\frac{9 x + 5}{6 - x}) + (6 - x) \cdot 9}$

Cancel the common factor of $6 - x$ .

Cancel the common factor.

$f (\frac{9 x + 5}{6 - x}) = \frac{(6 - x) (6 (\frac{9 x + 5}{6 - x})) + (6 - x) \cdot - 5}{(6 - x) (\frac{9 x + 5}{6 - x}) + (6 - x) \cdot 9}$

Rewrite the expression.

$f (\frac{9 x + 5}{6 - x}) = \frac{(6 - x) (6 (\frac{9 x + 5}{6 - x})) + (6 - x) \cdot - 5}{9 x + 5 + (6 - x) \cdot 9}$

Simplify the numerator.

Factor $6 - x$ out of $(6 - x) \cdot 6 \frac{9 x + 5}{6 - x} + (6 - x) \cdot - 5$ .

Factor $6 - x$ out of $(6 - x) \cdot 6 \frac{9 x + 5}{6 - x}$ .

$f (\frac{9 x + 5}{6 - x}) = \frac{(6 - x) (6 (\frac{9 x + 5}{6 - x})) + (6 - x) \cdot - 5}{9 x + 5 + (6 - x) \cdot 9}$

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

$f (\frac{9 x + 5}{6 - x}) = \frac{(6 - x) (6 (\frac{9 x + 5}{6 - x})) + (6 - x) (- 5)}{9 x + 5 + (6 - x) \cdot 9}$

Factor $6 - x$ out of $(6 - x) (6 \frac{9 x + 5}{6 - x}) + (6 - x) (- 5)$ .

$f (\frac{9 x + 5}{6 - x}) = \frac{(6 - x) (6 (\frac{9 x + 5}{6 - x}) - 5)}{9 x + 5 + (6 - x) \cdot 9}$

Combine $6$ and $\frac{9 x + 5}{6 - x}$ .

$f (\frac{9 x + 5}{6 - x}) = \frac{(6 - x) (\frac{6 (9 x + 5)}{6 - x} - 5)}{9 x + 5 + (6 - x) \cdot 9}$

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

$f (\frac{9 x + 5}{6 - x}) = \frac{(6 - x) (\frac{6 (9 x + 5)}{6 - x} - 5 \cdot \frac{6 - x}{6 - x})}{9 x + 5 + (6 - x) \cdot 9}$

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

$f (\frac{9 x + 5}{6 - x}) = \frac{(6 - x) (\frac{6 (9 x + 5)}{6 - x} + \frac{- 5 (6 - x)}{6 - x})}{9 x + 5 + (6 - x) \cdot 9}$

Combine the numerators over the common denominator.

$f (\frac{9 x + 5}{6 - x}) = \frac{(6 - x) (\frac{6 (9 x + 5) - 5 (6 - x)}{6 - x})}{9 x + 5 + (6 - x) \cdot 9}$

Reorder terms.

$f (\frac{9 x + 5}{6 - x}) = \frac{(6 - x) (\frac{6 (9 x + 5) - 5 (6 - x)}{- x + 6})}{9 x + 5 + (6 - x) \cdot 9}$

Rewrite $\frac{6 (9 x + 5) - 5 (6 - x)}{- x + 6}$ in a factored form.

Apply the distributive property.

$f (\frac{9 x + 5}{6 - x}) = \frac{(6 - x) (\frac{6 (9 x) + 6 \cdot 5 - 5 (6 - x)}{- x + 6})}{9 x + 5 + (6 - x) \cdot 9}$

Multiply $9$ by $6$ .

$f (\frac{9 x + 5}{6 - x}) = \frac{(6 - x) (\frac{54 x + 6 \cdot 5 - 5 (6 - x)}{- x + 6})}{9 x + 5 + (6 - x) \cdot 9}$

Multiply $6$ by $5$ .

$f (\frac{9 x + 5}{6 - x}) = \frac{(6 - x) (\frac{54 x + 30 - 5 (6 - x)}{- x + 6})}{9 x + 5 + (6 - x) \cdot 9}$

Apply the distributive property.

$f (\frac{9 x + 5}{6 - x}) = \frac{(6 - x) (\frac{54 x + 30 - 5 \cdot 6 - 5 (- x)}{- x + 6})}{9 x + 5 + (6 - x) \cdot 9}$

Multiply $- 5$ by $6$ .

$f (\frac{9 x + 5}{6 - x}) = \frac{(6 - x) (\frac{54 x + 30 - 30 - 5 (- x)}{- x + 6})}{9 x + 5 + (6 - x) \cdot 9}$

Multiply $- 1$ by $- 5$ .

$f (\frac{9 x + 5}{6 - x}) = \frac{(6 - x) (\frac{54 x + 30 - 30 + 5 x}{- x + 6})}{9 x + 5 + (6 - x) \cdot 9}$

Add $54 x$ and $5 x$ .

$f (\frac{9 x + 5}{6 - x}) = \frac{(6 - x) (\frac{59 x + 30 - 30}{- x + 6})}{9 x + 5 + (6 - x) \cdot 9}$

Subtract $30$ from $30$ .

$f (\frac{9 x + 5}{6 - x}) = \frac{(6 - x) (\frac{59 x + 0}{- x + 6})}{9 x + 5 + (6 - x) \cdot 9}$

Add $59 x$ and $0$ .

$f (\frac{9 x + 5}{6 - x}) = \frac{(6 - x) (\frac{59 x}{- x + 6})}{9 x + 5 + (6 - x) \cdot 9}$

Simplify the denominator.

Apply the distributive property.

$f (\frac{9 x + 5}{6 - x}) = \frac{(6 - x) (\frac{59 x}{- x + 6})}{9 x + 5 + 6 \cdot 9 - x \cdot 9}$

Multiply $6$ by $9$ .

$f (\frac{9 x + 5}{6 - x}) = \frac{(6 - x) (\frac{59 x}{- x + 6})}{9 x + 5 + 54 - x \cdot 9}$

Multiply $9$ by $- 1$ .

$f (\frac{9 x + 5}{6 - x}) = \frac{(6 - x) (\frac{59 x}{- x + 6})}{9 x + 5 + 54 - 9 x}$

Subtract $9 x$ from $9 x$ .

$f (\frac{9 x + 5}{6 - x}) = \frac{(6 - x) (\frac{59 x}{- x + 6})}{0 + 5 + 54}$

Add $0$ and $5$ .

$f (\frac{9 x + 5}{6 - x}) = \frac{(6 - x) (\frac{59 x}{- x + 6})}{5 + 54}$

Add $5$ and $54$ .

$f (\frac{9 x + 5}{6 - x}) = \frac{(6 - x) (\frac{59 x}{- x + 6})}{59}$

Simplify terms.

Factor $\frac{59 x}{- x + 6}$ out of $\frac{(6 - x) \frac{59 x}{- x + 6}}{59}$ .

$f (\frac{9 x + 5}{6 - x}) = \frac{59 x}{- x + 6} \cdot \frac{6 - x}{59}$

Cancel the common factor of $59$ .

Factor $59$ out of $59 x$ .

$f (\frac{9 x + 5}{6 - x}) = \frac{59 (x)}{- x + 6} \cdot \frac{6 - x}{59}$

Cancel the common factor.

$f (\frac{9 x + 5}{6 - x}) = \frac{59 x}{- x + 6} \cdot \frac{6 - x}{59}$

Rewrite the expression.

$f (\frac{9 x + 5}{6 - x}) = \frac{x}{- x + 6} \cdot (6 - x)$

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

$f (\frac{9 x + 5}{6 - x}) = \frac{x (6 - x)}{- x + 6}$

Cancel the common factor of $6 - x$ and $- x + 6$ .

Reorder terms.

$f (\frac{9 x + 5}{6 - x}) = \frac{x (- x + 6)}{- x + 6}$

Cancel the common factor.

$f (\frac{9 x + 5}{6 - x}) = \frac{x (- x + 6)}{- x + 6}$

Divide $x$ by $1$ .

$f (\frac{9 x + 5}{6 - x}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{9 x + 5}{6 - x}$ is the inverse of $f (x) = \frac{6 x - 5}{x + 9}$ .

$f^{- 1} (x) = \frac{9 x + 5}{6 - x}$

Find the Inverse (6x-5)/(x+9)

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