Find the Inverse (9x)/(2x-9)

$\frac{9 x}{2 x - 9}$

Step 1

Interchange the variables.

$x = \frac{9 y}{2 y - 9}$

Step 2

Solve for $y$ .

Multiply the equation by $2 y - 9$ .

$(2 y - 9) (x) = (\frac{9 y}{2 y - 9}) (2 y - 9)$

Simplify the left side.

Apply the distributive property.

$2 y x - 9 x = (\frac{9 y}{2 y - 9}) (2 y - 9)$

Simplify the right side.

Cancel the common factor of $2 y - 9$ .

Cancel the common factor.

$2 y x - 9 x = \frac{9 y}{2 y - 9} (2 y - 9)$

Rewrite the expression.

$2 y x - 9 x = 9 y$

Solve for $y$ .

Subtract $9 y$ from both sides of the equation.

$2 y x - 9 x - 9 y = 0$

Add $9 x$ to both sides of the equation.

$2 y x - 9 y = 9 x$

Factor $y$ out of $2 y x - 9 y$ .

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

$y (2 x) - 9 y = 9 x$

Factor $y$ out of $- 9 y$ .

$y (2 x) + y \cdot - 9 = 9 x$

Factor $y$ out of $y (2 x) + y \cdot - 9$ .

$y (2 x - 9) = 9 x$

Divide each term in $y (2 x - 9) = 9 x$ by $2 x - 9$ and simplify.

Divide each term in $y (2 x - 9) = 9 x$ by $2 x - 9$ .

$\frac{y (2 x - 9)}{2 x - 9} = \frac{9 x}{2 x - 9}$

Simplify the left side.

Cancel the common factor of $2 x - 9$ .

Cancel the common factor.

$\frac{y (2 x - 9)}{2 x - 9} = \frac{9 x}{2 x - 9}$

Divide $y$ by $1$ .

$y = \frac{9 x}{2 x - 9}$

Step 3

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

$f^{- 1} (x) = \frac{9 x}{2 x - 9}$

Step 4

Verify if $f^{- 1} (x) = \frac{9 x}{2 x - 9}$ is the inverse of $f (x) = \frac{9 x}{2 x - 9}$ .

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 $f^{- 1} (\frac{9 x}{2 x - 9})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{9 x}{2 x - 9}) = \frac{9 (\frac{9 x}{2 x - 9})}{2 (\frac{9 x}{2 x - 9}) - 9}$

Combine $9$ and $\frac{9 x}{2 x - 9}$ .

$f^{- 1} (\frac{9 x}{2 x - 9}) = \frac{\frac{9 (9 x)}{2 x - 9}}{2 (\frac{9 x}{2 x - 9}) - 9}$

Simplify the denominator.

Multiply $2 \frac{9 x}{2 x - 9}$ .

Combine $2$ and $\frac{9 x}{2 x - 9}$ .

$f^{- 1} (\frac{9 x}{2 x - 9}) = \frac{\frac{9 (9 x)}{2 x - 9}}{\frac{2 (9 x)}{2 x - 9} - 9}$

Multiply $9$ by $2$ .

$f^{- 1} (\frac{9 x}{2 x - 9}) = \frac{\frac{9 (9 x)}{2 x - 9}}{\frac{18 x}{2 x - 9} - 9}$

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{2 x - 9}{2 x - 9}$ .

$f^{- 1} (\frac{9 x}{2 x - 9}) = \frac{\frac{9 (9 x)}{2 x - 9}}{\frac{18 x}{2 x - 9} - 9 \cdot \frac{2 x - 9}{2 x - 9}}$

Combine $- 9$ and $\frac{2 x - 9}{2 x - 9}$ .

$f^{- 1} (\frac{9 x}{2 x - 9}) = \frac{\frac{9 (9 x)}{2 x - 9}}{\frac{18 x}{2 x - 9} + \frac{- 9 (2 x - 9)}{2 x - 9}}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{9 x}{2 x - 9}) = \frac{\frac{9 (9 x)}{2 x - 9}}{\frac{18 x - 9 (2 x - 9)}{2 x - 9}}$

Rewrite $\frac{18 x - 9 (2 x - 9)}{2 x - 9}$ in a factored form.

Factor $9$ out of $18 x - 9 (2 x - 9)$ .

Factor $9$ out of $18 x$ .

$f^{- 1} (\frac{9 x}{2 x - 9}) = \frac{\frac{9 (9 x)}{2 x - 9}}{\frac{9 (2 x) - 9 (2 x - 9)}{2 x - 9}}$

Factor $9$ out of $- 9 (2 x - 9)$ .

$f^{- 1} (\frac{9 x}{2 x - 9}) = \frac{\frac{9 (9 x)}{2 x - 9}}{\frac{9 (2 x) + 9 (- (2 x - 9))}{2 x - 9}}$

Factor $9$ out of $9 (2 x) + 9 (- (2 x - 9))$ .

$f^{- 1} (\frac{9 x}{2 x - 9}) = \frac{\frac{9 (9 x)}{2 x - 9}}{\frac{9 (2 x - (2 x - 9))}{2 x - 9}}$

Apply the distributive property.

$f^{- 1} (\frac{9 x}{2 x - 9}) = \frac{\frac{9 (9 x)}{2 x - 9}}{\frac{9 (2 x - (2 x) + 9)}{2 x - 9}}$

Multiply $2$ by $- 1$ .

$f^{- 1} (\frac{9 x}{2 x - 9}) = \frac{\frac{9 (9 x)}{2 x - 9}}{\frac{9 (2 x - 2 x + 9)}{2 x - 9}}$

Multiply $- 1$ by $- 9$ .

$f^{- 1} (\frac{9 x}{2 x - 9}) = \frac{\frac{9 (9 x)}{2 x - 9}}{\frac{9 (2 x - 2 x + 9)}{2 x - 9}}$

Subtract $2 x$ from $2 x$ .

$f^{- 1} (\frac{9 x}{2 x - 9}) = \frac{\frac{9 (9 x)}{2 x - 9}}{\frac{9 (0 + 9)}{2 x - 9}}$

Add $0$ and $9$ .

$f^{- 1} (\frac{9 x}{2 x - 9}) = \frac{\frac{9 (9 x)}{2 x - 9}}{\frac{9 \cdot 9}{2 x - 9}}$

Multiply $9$ by $9$ .

$f^{- 1} (\frac{9 x}{2 x - 9}) = \frac{\frac{9 (9 x)}{2 x - 9}}{\frac{81}{2 x - 9}}$

Multiply $9$ by $9$ .

$f^{- 1} (\frac{9 x}{2 x - 9}) = \frac{\frac{81 x}{2 x - 9}}{\frac{81}{2 x - 9}}$

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{9 x}{2 x - 9}) = \frac{81 x}{2 x - 9} \cdot \frac{2 x - 9}{81}$

Cancel the common factor of $81$ .

Factor $81$ out of $81 x$ .

$f^{- 1} (\frac{9 x}{2 x - 9}) = \frac{81 (x)}{2 x - 9} \cdot \frac{2 x - 9}{81}$

Cancel the common factor.

$f^{- 1} (\frac{9 x}{2 x - 9}) = \frac{81 x}{2 x - 9} \cdot \frac{2 x - 9}{81}$

Rewrite the expression.

$f^{- 1} (\frac{9 x}{2 x - 9}) = \frac{x}{2 x - 9} \cdot (2 x - 9)$

Cancel the common factor of $2 x - 9$ .

Cancel the common factor.

$f^{- 1} (\frac{9 x}{2 x - 9}) = \frac{x}{2 x - 9} \cdot (2 x - 9)$

Rewrite the expression.

$f^{- 1} (\frac{9 x}{2 x - 9}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{9 x}{2 x - 9})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{9 x}{2 x - 9}) = \frac{9 (\frac{9 x}{2 x - 9})}{2 (\frac{9 x}{2 x - 9}) - 9}$

Combine $9$ and $\frac{9 x}{2 x - 9}$ .

$f (\frac{9 x}{2 x - 9}) = \frac{\frac{9 (9 x)}{2 x - 9}}{2 (\frac{9 x}{2 x - 9}) - 9}$

Simplify the denominator.

Multiply $2 \frac{9 x}{2 x - 9}$ .

Combine $2$ and $\frac{9 x}{2 x - 9}$ .

$f (\frac{9 x}{2 x - 9}) = \frac{\frac{9 (9 x)}{2 x - 9}}{\frac{2 (9 x)}{2 x - 9} - 9}$

Multiply $9$ by $2$ .

$f (\frac{9 x}{2 x - 9}) = \frac{\frac{9 (9 x)}{2 x - 9}}{\frac{18 x}{2 x - 9} - 9}$

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{2 x - 9}{2 x - 9}$ .

$f (\frac{9 x}{2 x - 9}) = \frac{\frac{9 (9 x)}{2 x - 9}}{\frac{18 x}{2 x - 9} - 9 \cdot \frac{2 x - 9}{2 x - 9}}$

Combine $- 9$ and $\frac{2 x - 9}{2 x - 9}$ .

$f (\frac{9 x}{2 x - 9}) = \frac{\frac{9 (9 x)}{2 x - 9}}{\frac{18 x}{2 x - 9} + \frac{- 9 (2 x - 9)}{2 x - 9}}$

Combine the numerators over the common denominator.

$f (\frac{9 x}{2 x - 9}) = \frac{\frac{9 (9 x)}{2 x - 9}}{\frac{18 x - 9 (2 x - 9)}{2 x - 9}}$

Rewrite $\frac{18 x - 9 (2 x - 9)}{2 x - 9}$ in a factored form.

Factor $9$ out of $18 x - 9 (2 x - 9)$ .

Factor $9$ out of $18 x$ .

$f (\frac{9 x}{2 x - 9}) = \frac{\frac{9 (9 x)}{2 x - 9}}{\frac{9 (2 x) - 9 (2 x - 9)}{2 x - 9}}$

Factor $9$ out of $- 9 (2 x - 9)$ .

$f (\frac{9 x}{2 x - 9}) = \frac{\frac{9 (9 x)}{2 x - 9}}{\frac{9 (2 x) + 9 (- (2 x - 9))}{2 x - 9}}$

Factor $9$ out of $9 (2 x) + 9 (- (2 x - 9))$ .

$f (\frac{9 x}{2 x - 9}) = \frac{\frac{9 (9 x)}{2 x - 9}}{\frac{9 (2 x - (2 x - 9))}{2 x - 9}}$

Apply the distributive property.

$f (\frac{9 x}{2 x - 9}) = \frac{\frac{9 (9 x)}{2 x - 9}}{\frac{9 (2 x - (2 x) + 9)}{2 x - 9}}$

Multiply $2$ by $- 1$ .

$f (\frac{9 x}{2 x - 9}) = \frac{\frac{9 (9 x)}{2 x - 9}}{\frac{9 (2 x - 2 x + 9)}{2 x - 9}}$

Multiply $- 1$ by $- 9$ .

$f (\frac{9 x}{2 x - 9}) = \frac{\frac{9 (9 x)}{2 x - 9}}{\frac{9 (2 x - 2 x + 9)}{2 x - 9}}$

Subtract $2 x$ from $2 x$ .

$f (\frac{9 x}{2 x - 9}) = \frac{\frac{9 (9 x)}{2 x - 9}}{\frac{9 (0 + 9)}{2 x - 9}}$

Add $0$ and $9$ .

$f (\frac{9 x}{2 x - 9}) = \frac{\frac{9 (9 x)}{2 x - 9}}{\frac{9 \cdot 9}{2 x - 9}}$

Multiply $9$ by $9$ .

$f (\frac{9 x}{2 x - 9}) = \frac{\frac{9 (9 x)}{2 x - 9}}{\frac{81}{2 x - 9}}$

Multiply $9$ by $9$ .

$f (\frac{9 x}{2 x - 9}) = \frac{\frac{81 x}{2 x - 9}}{\frac{81}{2 x - 9}}$

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{9 x}{2 x - 9}) = \frac{81 x}{2 x - 9} \cdot \frac{2 x - 9}{81}$

Cancel the common factor of $81$ .

Factor $81$ out of $81 x$ .

$f (\frac{9 x}{2 x - 9}) = \frac{81 (x)}{2 x - 9} \cdot \frac{2 x - 9}{81}$

Cancel the common factor.

$f (\frac{9 x}{2 x - 9}) = \frac{81 x}{2 x - 9} \cdot \frac{2 x - 9}{81}$

Rewrite the expression.

$f (\frac{9 x}{2 x - 9}) = \frac{x}{2 x - 9} \cdot (2 x - 9)$

Cancel the common factor of $2 x - 9$ .

Cancel the common factor.

$f (\frac{9 x}{2 x - 9}) = \frac{x}{2 x - 9} \cdot (2 x - 9)$

Rewrite the expression.

$f (\frac{9 x}{2 x - 9}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{9 x}{2 x - 9}$ is the inverse of $f (x) = \frac{9 x}{2 x - 9}$ .

$f^{- 1} (x) = \frac{9 x}{2 x - 9}$

Find the Inverse (9x)/(2x-9)

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