Find the Inverse (7x)/(9x-1)

$\frac{7 x}{9 x - 1}$

Step 1

Interchange the variables.

$x = \frac{7 y}{9 y - 1}$

Step 2

Solve for $y$ .

Multiply the equation by $9 y - 1$ .

$(9 y - 1) (x) = (\frac{7 y}{9 y - 1}) (9 y - 1)$

Simplify the left side.

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

Apply the distributive property.

$9 y x - 1 x = (\frac{7 y}{9 y - 1}) (9 y - 1)$

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

$9 y x - x = (\frac{7 y}{9 y - 1}) (9 y - 1)$

Simplify the right side.

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

Cancel the common factor.

$9 y x - x = \frac{7 y}{9 y - 1} (9 y - 1)$

Rewrite the expression.

$9 y x - x = 7 y$

Solve for $y$ .

Subtract $7 y$ from both sides of the equation.

$9 y x - x - 7 y = 0$

Add $x$ to both sides of the equation.

$9 y x - 7 y = x$

Factor $y$ out of $9 y x - 7 y$ .

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

$y (9 x) - 7 y = x$

Factor $y$ out of $- 7 y$ .

$y (9 x) + y \cdot - 7 = x$

Factor $y$ out of $y (9 x) + y \cdot - 7$ .

$y (9 x - 7) = x$

Divide each term in $y (9 x - 7) = x$ by $9 x - 7$ and simplify.

Divide each term in $y (9 x - 7) = x$ by $9 x - 7$ .

$\frac{y (9 x - 7)}{9 x - 7} = \frac{x}{9 x - 7}$

Simplify the left side.

Cancel the common factor of $9 x - 7$ .

Cancel the common factor.

$\frac{y (9 x - 7)}{9 x - 7} = \frac{x}{9 x - 7}$

Divide $y$ by $1$ .

$y = \frac{x}{9 x - 7}$

Step 3

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

$f^{- 1} (x) = \frac{x}{9 x - 7}$

Step 4

Verify if $f^{- 1} (x) = \frac{x}{9 x - 7}$ is the inverse of $f (x) = \frac{7 x}{9 x - 1}$ .

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 $f^{- 1} (\frac{7 x}{9 x - 1})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{7 x}{9 x - 1}) = \frac{\frac{7 x}{9 x - 1}}{9 (\frac{7 x}{9 x - 1}) - 7}$

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{7 x}{9 x - 1}) = \frac{7 x}{9 x - 1} \cdot \frac{1}{9 (\frac{7 x}{9 x - 1}) - 7}$

Simplify the denominator.

Multiply $9 \frac{7 x}{9 x - 1}$ .

Combine $9$ and $\frac{7 x}{9 x - 1}$ .

$f^{- 1} (\frac{7 x}{9 x - 1}) = \frac{7 x}{9 x - 1} \cdot \frac{1}{\frac{9 (7 x)}{9 x - 1} - 7}$

Multiply $7$ by $9$ .

$f^{- 1} (\frac{7 x}{9 x - 1}) = \frac{7 x}{9 x - 1} \cdot \frac{1}{\frac{63 x}{9 x - 1} - 7}$

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{9 x - 1}{9 x - 1}$ .

$f^{- 1} (\frac{7 x}{9 x - 1}) = \frac{7 x}{9 x - 1} \cdot \frac{1}{\frac{63 x}{9 x - 1} - 7 \cdot \frac{9 x - 1}{9 x - 1}}$

Combine $- 7$ and $\frac{9 x - 1}{9 x - 1}$ .

$f^{- 1} (\frac{7 x}{9 x - 1}) = \frac{7 x}{9 x - 1} \cdot \frac{1}{\frac{63 x}{9 x - 1} + \frac{- 7 (9 x - 1)}{9 x - 1}}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{7 x}{9 x - 1}) = \frac{7 x}{9 x - 1} \cdot \frac{1}{\frac{63 x - 7 (9 x - 1)}{9 x - 1}}$

Rewrite $\frac{63 x - 7 (9 x - 1)}{9 x - 1}$ in a factored form.

Factor $7$ out of $63 x - 7 (9 x - 1)$ .

Factor $7$ out of $63 x$ .

$f^{- 1} (\frac{7 x}{9 x - 1}) = \frac{7 x}{9 x - 1} \cdot \frac{1}{\frac{7 (9 x) - 7 (9 x - 1)}{9 x - 1}}$

Factor $7$ out of $- 7 (9 x - 1)$ .

$f^{- 1} (\frac{7 x}{9 x - 1}) = \frac{7 x}{9 x - 1} \cdot \frac{1}{\frac{7 (9 x) + 7 (- (9 x - 1))}{9 x - 1}}$

Factor $7$ out of $7 (9 x) + 7 (- (9 x - 1))$ .

$f^{- 1} (\frac{7 x}{9 x - 1}) = \frac{7 x}{9 x - 1} \cdot \frac{1}{\frac{7 (9 x - (9 x - 1))}{9 x - 1}}$

Apply the distributive property.

$f^{- 1} (\frac{7 x}{9 x - 1}) = \frac{7 x}{9 x - 1} \cdot \frac{1}{\frac{7 (9 x - (9 x) + 1)}{9 x - 1}}$

Multiply $9$ by $- 1$ .

$f^{- 1} (\frac{7 x}{9 x - 1}) = \frac{7 x}{9 x - 1} \cdot \frac{1}{\frac{7 (9 x - 9 x + 1)}{9 x - 1}}$

Multiply $- 1$ by $- 1$ .

$f^{- 1} (\frac{7 x}{9 x - 1}) = \frac{7 x}{9 x - 1} \cdot \frac{1}{\frac{7 (9 x - 9 x + 1)}{9 x - 1}}$

Subtract $9 x$ from $9 x$ .

$f^{- 1} (\frac{7 x}{9 x - 1}) = \frac{7 x}{9 x - 1} \cdot \frac{1}{\frac{7 (0 + 1)}{9 x - 1}}$

Add $0$ and $1$ .

$f^{- 1} (\frac{7 x}{9 x - 1}) = \frac{7 x}{9 x - 1} \cdot \frac{1}{\frac{7 \cdot 1}{9 x - 1}}$

Multiply $7$ by $1$ .

$f^{- 1} (\frac{7 x}{9 x - 1}) = \frac{7 x}{9 x - 1} \cdot \frac{1}{\frac{7}{9 x - 1}}$

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{7 x}{9 x - 1}) = \frac{7 x}{9 x - 1} \cdot (1 (\frac{9 x - 1}{7}))$

Multiply $\frac{9 x - 1}{7}$ by $1$ .

$f^{- 1} (\frac{7 x}{9 x - 1}) = \frac{7 x}{9 x - 1} \cdot \frac{9 x - 1}{7}$

Cancel the common factor of $7$ .

Factor $7$ out of $7 x$ .

$f^{- 1} (\frac{7 x}{9 x - 1}) = \frac{7 (x)}{9 x - 1} \cdot \frac{9 x - 1}{7}$

Cancel the common factor.

$f^{- 1} (\frac{7 x}{9 x - 1}) = \frac{7 x}{9 x - 1} \cdot \frac{9 x - 1}{7}$

Rewrite the expression.

$f^{- 1} (\frac{7 x}{9 x - 1}) = \frac{x}{9 x - 1} \cdot (9 x - 1)$

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

Cancel the common factor.

$f^{- 1} (\frac{7 x}{9 x - 1}) = \frac{x}{9 x - 1} \cdot (9 x - 1)$

Rewrite the expression.

$f^{- 1} (\frac{7 x}{9 x - 1}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{x}{9 x - 7})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x}{9 x - 7}) = \frac{7 (\frac{x}{9 x - 7})}{9 (\frac{x}{9 x - 7}) - 1}$

Combine $7$ and $\frac{x}{9 x - 7}$ .

$f (\frac{x}{9 x - 7}) = \frac{\frac{7 x}{9 x - 7}}{9 (\frac{x}{9 x - 7}) - 1}$

Simplify the denominator.

Combine $9$ and $\frac{x}{9 x - 7}$ .

$f (\frac{x}{9 x - 7}) = \frac{\frac{7 x}{9 x - 7}}{\frac{9 x}{9 x - 7} - 1}$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{9 x - 7}{9 x - 7}$ .

$f (\frac{x}{9 x - 7}) = \frac{\frac{7 x}{9 x - 7}}{\frac{9 x}{9 x - 7} - 1 \cdot \frac{9 x - 7}{9 x - 7}}$

Combine $- 1$ and $\frac{9 x - 7}{9 x - 7}$ .

$f (\frac{x}{9 x - 7}) = \frac{\frac{7 x}{9 x - 7}}{\frac{9 x}{9 x - 7} + \frac{- (9 x - 7)}{9 x - 7}}$

Combine the numerators over the common denominator.

$f (\frac{x}{9 x - 7}) = \frac{\frac{7 x}{9 x - 7}}{\frac{9 x - (9 x - 7)}{9 x - 7}}$

Rewrite $\frac{9 x - (9 x - 7)}{9 x - 7}$ in a factored form.

Apply the distributive property.

$f (\frac{x}{9 x - 7}) = \frac{\frac{7 x}{9 x - 7}}{\frac{9 x - (9 x) + 7}{9 x - 7}}$

Multiply $9$ by $- 1$ .

$f (\frac{x}{9 x - 7}) = \frac{\frac{7 x}{9 x - 7}}{\frac{9 x - 9 x + 7}{9 x - 7}}$

Multiply $- 1$ by $- 7$ .

$f (\frac{x}{9 x - 7}) = \frac{\frac{7 x}{9 x - 7}}{\frac{9 x - 9 x + 7}{9 x - 7}}$

Subtract $9 x$ from $9 x$ .

$f (\frac{x}{9 x - 7}) = \frac{\frac{7 x}{9 x - 7}}{\frac{0 + 7}{9 x - 7}}$

Add $0$ and $7$ .

$f (\frac{x}{9 x - 7}) = \frac{\frac{7 x}{9 x - 7}}{\frac{7}{9 x - 7}}$

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{x}{9 x - 7}) = \frac{7 x}{9 x - 7} \cdot \frac{9 x - 7}{7}$

Cancel the common factor of $7$ .

Factor $7$ out of $7 x$ .

$f (\frac{x}{9 x - 7}) = \frac{7 (x)}{9 x - 7} \cdot \frac{9 x - 7}{7}$

Cancel the common factor.

$f (\frac{x}{9 x - 7}) = \frac{7 x}{9 x - 7} \cdot \frac{9 x - 7}{7}$

Rewrite the expression.

$f (\frac{x}{9 x - 7}) = \frac{x}{9 x - 7} \cdot (9 x - 7)$

Cancel the common factor of $9 x - 7$ .

Cancel the common factor.

$f (\frac{x}{9 x - 7}) = \frac{x}{9 x - 7} \cdot (9 x - 7)$

Rewrite the expression.

$f (\frac{x}{9 x - 7}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x}{9 x - 7}$ is the inverse of $f (x) = \frac{7 x}{9 x - 1}$ .

$f^{- 1} (x) = \frac{x}{9 x - 7}$

Find the Inverse (7x)/(9x-1)

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