Find the Inverse (2x)/(9x-1)

$\frac{2 x}{9 x - 1}$

Step 1

Interchange the variables.

$x = \frac{2 y}{9 y - 1}$

Step 2

Solve for $y$ .

Multiply the equation by $9 y - 1$ .

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

Simplify the left side.

Simplify $(9 y - 1) (x)$ .

Apply the distributive property.

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

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

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

Simplify the right side.

Cancel the common factor of $9 y - 1$ .

Cancel the common factor.

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

Rewrite the expression.

$9 y x - x = 2 y$

Solve for $y$ .

Subtract $2 y$ from both sides of the equation.

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

Add $x$ to both sides of the equation.

$9 y x - 2 y = x$

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

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

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

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

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

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

$y (9 x - 2) = x$

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

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

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

Simplify the left side.

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

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Step 3

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

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

Step 4

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

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

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

Multiply the numerator by the reciprocal of the denominator.

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

Simplify the denominator.

Multiply $9 \frac{2 x}{9 x - 1}$ .

Combine $9$ and $\frac{2 x}{9 x - 1}$ .

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

Multiply $2$ by $9$ .

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

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{9 x - 1}{9 x - 1}$ .

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

Combine $- 2$ and $\frac{9 x - 1}{9 x - 1}$ .

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

Combine the numerators over the common denominator.

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

Rewrite $\frac{18 x - 2 (9 x - 1)}{9 x - 1}$ in a factored form.

Factor $2$ out of $18 x - 2 (9 x - 1)$ .

Factor $2$ out of $18 x$ .

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

Factor $2$ out of $- 2 (9 x - 1)$ .

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

Factor $2$ out of $2 (9 x) + 2 (- (9 x - 1))$ .

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

Apply the distributive property.

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

Multiply $9$ by $- 1$ .

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

Multiply $- 1$ by $- 1$ .

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

Subtract $9 x$ from $9 x$ .

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

Add $0$ and $1$ .

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

Multiply $2$ by $1$ .

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

Multiply the numerator by the reciprocal of the denominator.

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

Multiply $\frac{9 x - 1}{2}$ by $1$ .

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

Cancel the common factor of $2$ .

Factor $2$ out of $2 x$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $9 x - 1$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

Set up the composite result function.

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

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

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

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

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

Simplify the denominator.

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

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

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{9 x - 2}{9 x - 2}$ .

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

Combine $- 1$ and $\frac{9 x - 2}{9 x - 2}$ .

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

Combine the numerators over the common denominator.

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

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

Apply the distributive property.

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

Multiply $9$ by $- 1$ .

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

Multiply $- 1$ by $- 2$ .

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

Subtract $9 x$ from $9 x$ .

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

Add $0$ and $2$ .

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

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $2$ .

Factor $2$ out of $2 x$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Find the Inverse (2x)/(9x-1)

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