Find the Inverse (4x)/(9x-7)

$\frac{4 x}{9 x - 7}$

Step 1

Interchange the variables.

$x = \frac{4 y}{9 y - 7}$

Step 2

Solve for $y$ .

Multiply the equation by $9 y - 7$ .

$(9 y - 7) (x) = (\frac{4 y}{9 y - 7}) (9 y - 7)$

Simplify the left side.

Apply the distributive property.

$9 y x - 7 x = (\frac{4 y}{9 y - 7}) (9 y - 7)$

Simplify the right side.

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

Cancel the common factor.

$9 y x - 7 x = \frac{4 y}{9 y - 7} (9 y - 7)$

Rewrite the expression.

$9 y x - 7 x = 4 y$

Solve for $y$ .

Subtract $4 y$ from both sides of the equation.

$9 y x - 7 x - 4 y = 0$

Add $7 x$ to both sides of the equation.

$9 y x - 4 y = 7 x$

Factor $y$ out of $9 y x - 4 y$ .

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

$y (9 x) - 4 y = 7 x$

Factor $y$ out of $- 4 y$ .

$y (9 x) + y \cdot - 4 = 7 x$

Factor $y$ out of $y (9 x) + y \cdot - 4$ .

$y (9 x - 4) = 7 x$

Divide each term in $y (9 x - 4) = 7 x$ by $9 x - 4$ and simplify.

Divide each term in $y (9 x - 4) = 7 x$ by $9 x - 4$ .

$\frac{y (9 x - 4)}{9 x - 4} = \frac{7 x}{9 x - 4}$

Simplify the left side.

Cancel the common factor of $9 x - 4$ .

Cancel the common factor.

$\frac{y (9 x - 4)}{9 x - 4} = \frac{7 x}{9 x - 4}$

Divide $y$ by $1$ .

$y = \frac{7 x}{9 x - 4}$

Step 3

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

$f^{- 1} (x) = \frac{7 x}{9 x - 4}$

Step 4

Verify if $f^{- 1} (x) = \frac{7 x}{9 x - 4}$ is the inverse of $f (x) = \frac{4 x}{9 x - 7}$ .

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 $f^{- 1} (\frac{4 x}{9 x - 7})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{4 x}{9 x - 7}) = \frac{7 (\frac{4 x}{9 x - 7})}{9 (\frac{4 x}{9 x - 7}) - 4}$

Combine $7$ and $\frac{4 x}{9 x - 7}$ .

$f^{- 1} (\frac{4 x}{9 x - 7}) = \frac{\frac{7 (4 x)}{9 x - 7}}{9 (\frac{4 x}{9 x - 7}) - 4}$

Simplify the denominator.

Multiply $9 \frac{4 x}{9 x - 7}$ .

Combine $9$ and $\frac{4 x}{9 x - 7}$ .

$f^{- 1} (\frac{4 x}{9 x - 7}) = \frac{\frac{7 (4 x)}{9 x - 7}}{\frac{9 (4 x)}{9 x - 7} - 4}$

Multiply $4$ by $9$ .

$f^{- 1} (\frac{4 x}{9 x - 7}) = \frac{\frac{7 (4 x)}{9 x - 7}}{\frac{36 x}{9 x - 7} - 4}$

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{9 x - 7}{9 x - 7}$ .

$f^{- 1} (\frac{4 x}{9 x - 7}) = \frac{\frac{7 (4 x)}{9 x - 7}}{\frac{36 x}{9 x - 7} - 4 \cdot \frac{9 x - 7}{9 x - 7}}$

Combine $- 4$ and $\frac{9 x - 7}{9 x - 7}$ .

$f^{- 1} (\frac{4 x}{9 x - 7}) = \frac{\frac{7 (4 x)}{9 x - 7}}{\frac{36 x}{9 x - 7} + \frac{- 4 (9 x - 7)}{9 x - 7}}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{4 x}{9 x - 7}) = \frac{\frac{7 (4 x)}{9 x - 7}}{\frac{36 x - 4 (9 x - 7)}{9 x - 7}}$

Rewrite $\frac{36 x - 4 (9 x - 7)}{9 x - 7}$ in a factored form.

Factor $4$ out of $36 x - 4 (9 x - 7)$ .

Factor $4$ out of $36 x$ .

$f^{- 1} (\frac{4 x}{9 x - 7}) = \frac{\frac{7 (4 x)}{9 x - 7}}{\frac{4 (9 x) - 4 (9 x - 7)}{9 x - 7}}$

Factor $4$ out of $- 4 (9 x - 7)$ .

$f^{- 1} (\frac{4 x}{9 x - 7}) = \frac{\frac{7 (4 x)}{9 x - 7}}{\frac{4 (9 x) + 4 (- (9 x - 7))}{9 x - 7}}$

Factor $4$ out of $4 (9 x) + 4 (- (9 x - 7))$ .

$f^{- 1} (\frac{4 x}{9 x - 7}) = \frac{\frac{7 (4 x)}{9 x - 7}}{\frac{4 (9 x - (9 x - 7))}{9 x - 7}}$

Apply the distributive property.

$f^{- 1} (\frac{4 x}{9 x - 7}) = \frac{\frac{7 (4 x)}{9 x - 7}}{\frac{4 (9 x - (9 x) + 7)}{9 x - 7}}$

Multiply $9$ by $- 1$ .

$f^{- 1} (\frac{4 x}{9 x - 7}) = \frac{\frac{7 (4 x)}{9 x - 7}}{\frac{4 (9 x - 9 x + 7)}{9 x - 7}}$

Multiply $- 1$ by $- 7$ .

$f^{- 1} (\frac{4 x}{9 x - 7}) = \frac{\frac{7 (4 x)}{9 x - 7}}{\frac{4 (9 x - 9 x + 7)}{9 x - 7}}$

Subtract $9 x$ from $9 x$ .

$f^{- 1} (\frac{4 x}{9 x - 7}) = \frac{\frac{7 (4 x)}{9 x - 7}}{\frac{4 (0 + 7)}{9 x - 7}}$

Add $0$ and $7$ .

$f^{- 1} (\frac{4 x}{9 x - 7}) = \frac{\frac{7 (4 x)}{9 x - 7}}{\frac{4 \cdot 7}{9 x - 7}}$

Multiply $4$ by $7$ .

$f^{- 1} (\frac{4 x}{9 x - 7}) = \frac{\frac{7 (4 x)}{9 x - 7}}{\frac{28}{9 x - 7}}$

Multiply $7$ by $4$ .

$f^{- 1} (\frac{4 x}{9 x - 7}) = \frac{\frac{28 x}{9 x - 7}}{\frac{28}{9 x - 7}}$

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{4 x}{9 x - 7}) = \frac{28 x}{9 x - 7} \cdot \frac{9 x - 7}{28}$

Cancel the common factor of $28$ .

Factor $28$ out of $28 x$ .

$f^{- 1} (\frac{4 x}{9 x - 7}) = \frac{28 (x)}{9 x - 7} \cdot \frac{9 x - 7}{28}$

Cancel the common factor.

$f^{- 1} (\frac{4 x}{9 x - 7}) = \frac{28 x}{9 x - 7} \cdot \frac{9 x - 7}{28}$

Rewrite the expression.

$f^{- 1} (\frac{4 x}{9 x - 7}) = \frac{x}{9 x - 7} \cdot (9 x - 7)$

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

Cancel the common factor.

$f^{- 1} (\frac{4 x}{9 x - 7}) = \frac{x}{9 x - 7} \cdot (9 x - 7)$

Rewrite the expression.

$f^{- 1} (\frac{4 x}{9 x - 7}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{7 x}{9 x - 4})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{7 x}{9 x - 4}) = \frac{4 (\frac{7 x}{9 x - 4})}{9 (\frac{7 x}{9 x - 4}) - 7}$

Combine $4$ and $\frac{7 x}{9 x - 4}$ .

$f (\frac{7 x}{9 x - 4}) = \frac{\frac{4 (7 x)}{9 x - 4}}{9 (\frac{7 x}{9 x - 4}) - 7}$

Simplify the denominator.

Multiply $9 \frac{7 x}{9 x - 4}$ .

Combine $9$ and $\frac{7 x}{9 x - 4}$ .

$f (\frac{7 x}{9 x - 4}) = \frac{\frac{4 (7 x)}{9 x - 4}}{\frac{9 (7 x)}{9 x - 4} - 7}$

Multiply $7$ by $9$ .

$f (\frac{7 x}{9 x - 4}) = \frac{\frac{4 (7 x)}{9 x - 4}}{\frac{63 x}{9 x - 4} - 7}$

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{9 x - 4}{9 x - 4}$ .

$f (\frac{7 x}{9 x - 4}) = \frac{\frac{4 (7 x)}{9 x - 4}}{\frac{63 x}{9 x - 4} - 7 \cdot \frac{9 x - 4}{9 x - 4}}$

Combine $- 7$ and $\frac{9 x - 4}{9 x - 4}$ .

$f (\frac{7 x}{9 x - 4}) = \frac{\frac{4 (7 x)}{9 x - 4}}{\frac{63 x}{9 x - 4} + \frac{- 7 (9 x - 4)}{9 x - 4}}$

Combine the numerators over the common denominator.

$f (\frac{7 x}{9 x - 4}) = \frac{\frac{4 (7 x)}{9 x - 4}}{\frac{63 x - 7 (9 x - 4)}{9 x - 4}}$

Rewrite $\frac{63 x - 7 (9 x - 4)}{9 x - 4}$ in a factored form.

Factor $7$ out of $63 x - 7 (9 x - 4)$ .

Factor $7$ out of $63 x$ .

$f (\frac{7 x}{9 x - 4}) = \frac{\frac{4 (7 x)}{9 x - 4}}{\frac{7 (9 x) - 7 (9 x - 4)}{9 x - 4}}$

Factor $7$ out of $- 7 (9 x - 4)$ .

$f (\frac{7 x}{9 x - 4}) = \frac{\frac{4 (7 x)}{9 x - 4}}{\frac{7 (9 x) + 7 (- (9 x - 4))}{9 x - 4}}$

Factor $7$ out of $7 (9 x) + 7 (- (9 x - 4))$ .

$f (\frac{7 x}{9 x - 4}) = \frac{\frac{4 (7 x)}{9 x - 4}}{\frac{7 (9 x - (9 x - 4))}{9 x - 4}}$

Apply the distributive property.

$f (\frac{7 x}{9 x - 4}) = \frac{\frac{4 (7 x)}{9 x - 4}}{\frac{7 (9 x - (9 x) + 4)}{9 x - 4}}$

Multiply $9$ by $- 1$ .

$f (\frac{7 x}{9 x - 4}) = \frac{\frac{4 (7 x)}{9 x - 4}}{\frac{7 (9 x - 9 x + 4)}{9 x - 4}}$

Multiply $- 1$ by $- 4$ .

$f (\frac{7 x}{9 x - 4}) = \frac{\frac{4 (7 x)}{9 x - 4}}{\frac{7 (9 x - 9 x + 4)}{9 x - 4}}$

Subtract $9 x$ from $9 x$ .

$f (\frac{7 x}{9 x - 4}) = \frac{\frac{4 (7 x)}{9 x - 4}}{\frac{7 (0 + 4)}{9 x - 4}}$

Add $0$ and $4$ .

$f (\frac{7 x}{9 x - 4}) = \frac{\frac{4 (7 x)}{9 x - 4}}{\frac{7 \cdot 4}{9 x - 4}}$

Multiply $7$ by $4$ .

$f (\frac{7 x}{9 x - 4}) = \frac{\frac{4 (7 x)}{9 x - 4}}{\frac{28}{9 x - 4}}$

Multiply $4$ by $7$ .

$f (\frac{7 x}{9 x - 4}) = \frac{\frac{28 x}{9 x - 4}}{\frac{28}{9 x - 4}}$

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{7 x}{9 x - 4}) = \frac{28 x}{9 x - 4} \cdot \frac{9 x - 4}{28}$

Cancel the common factor of $28$ .

Factor $28$ out of $28 x$ .

$f (\frac{7 x}{9 x - 4}) = \frac{28 (x)}{9 x - 4} \cdot \frac{9 x - 4}{28}$

Cancel the common factor.

$f (\frac{7 x}{9 x - 4}) = \frac{28 x}{9 x - 4} \cdot \frac{9 x - 4}{28}$

Rewrite the expression.

$f (\frac{7 x}{9 x - 4}) = \frac{x}{9 x - 4} \cdot (9 x - 4)$

Cancel the common factor of $9 x - 4$ .

Cancel the common factor.

$f (\frac{7 x}{9 x - 4}) = \frac{x}{9 x - 4} \cdot (9 x - 4)$

Rewrite the expression.

$f (\frac{7 x}{9 x - 4}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{7 x}{9 x - 4}$ is the inverse of $f (x) = \frac{4 x}{9 x - 7}$ .

$f^{- 1} (x) = \frac{7 x}{9 x - 4}$

Find the Inverse (4x)/(9x-7)

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