Find the Inverse (4x)/(7x-6)

$\frac{4 x}{7 x - 6}$

Step 1

Interchange the variables.

$x = \frac{4 y}{7 y - 6}$

Step 2

Solve for $y$ .

Multiply the equation by $7 y - 6$ .

$(7 y - 6) (x) = (\frac{4 y}{7 y - 6}) (7 y - 6)$

Simplify the left side.

Apply the distributive property.

$7 y x - 6 x = (\frac{4 y}{7 y - 6}) (7 y - 6)$

Simplify the right side.

Cancel the common factor of $7 y - 6$ .

Cancel the common factor.

$7 y x - 6 x = \frac{4 y}{7 y - 6} (7 y - 6)$

Rewrite the expression.

$7 y x - 6 x = 4 y$

Solve for $y$ .

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

$7 y x - 6 x - 4 y = 0$

Add $6 x$ to both sides of the equation.

$7 y x - 4 y = 6 x$

Factor $y$ out of $7 y x - 4 y$ .

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

$y (7 x) - 4 y = 6 x$

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

$y (7 x) + y \cdot - 4 = 6 x$

Factor $y$ out of $y (7 x) + y \cdot - 4$ .

$y (7 x - 4) = 6 x$

Divide each term in $y (7 x - 4) = 6 x$ by $7 x - 4$ and simplify.

Divide each term in $y (7 x - 4) = 6 x$ by $7 x - 4$ .

$\frac{y (7 x - 4)}{7 x - 4} = \frac{6 x}{7 x - 4}$

Simplify the left side.

Cancel the common factor of $7 x - 4$ .

Cancel the common factor.

$\frac{y (7 x - 4)}{7 x - 4} = \frac{6 x}{7 x - 4}$

Divide $y$ by $1$ .

$y = \frac{6 x}{7 x - 4}$

Step 3

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

$f^{- 1} (x) = \frac{6 x}{7 x - 4}$

Step 4

Verify if $f^{- 1} (x) = \frac{6 x}{7 x - 4}$ is the inverse of $f (x) = \frac{4 x}{7 x - 6}$ .

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 $f^{- 1} (\frac{4 x}{7 x - 6})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{4 x}{7 x - 6}) = \frac{6 (\frac{4 x}{7 x - 6})}{7 (\frac{4 x}{7 x - 6}) - 4}$

Combine $6$ and $\frac{4 x}{7 x - 6}$ .

$f^{- 1} (\frac{4 x}{7 x - 6}) = \frac{\frac{6 (4 x)}{7 x - 6}}{7 (\frac{4 x}{7 x - 6}) - 4}$

Simplify the denominator.

Multiply $7 \frac{4 x}{7 x - 6}$ .

Combine $7$ and $\frac{4 x}{7 x - 6}$ .

$f^{- 1} (\frac{4 x}{7 x - 6}) = \frac{\frac{6 (4 x)}{7 x - 6}}{\frac{7 (4 x)}{7 x - 6} - 4}$

Multiply $4$ by $7$ .

$f^{- 1} (\frac{4 x}{7 x - 6}) = \frac{\frac{6 (4 x)}{7 x - 6}}{\frac{28 x}{7 x - 6} - 4}$

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{7 x - 6}{7 x - 6}$ .

$f^{- 1} (\frac{4 x}{7 x - 6}) = \frac{\frac{6 (4 x)}{7 x - 6}}{\frac{28 x}{7 x - 6} - 4 \cdot \frac{7 x - 6}{7 x - 6}}$

Combine $- 4$ and $\frac{7 x - 6}{7 x - 6}$ .

$f^{- 1} (\frac{4 x}{7 x - 6}) = \frac{\frac{6 (4 x)}{7 x - 6}}{\frac{28 x}{7 x - 6} + \frac{- 4 (7 x - 6)}{7 x - 6}}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{4 x}{7 x - 6}) = \frac{\frac{6 (4 x)}{7 x - 6}}{\frac{28 x - 4 (7 x - 6)}{7 x - 6}}$

Rewrite $\frac{28 x - 4 (7 x - 6)}{7 x - 6}$ in a factored form.

Factor $4$ out of $28 x - 4 (7 x - 6)$ .

Factor $4$ out of $28 x$ .

$f^{- 1} (\frac{4 x}{7 x - 6}) = \frac{\frac{6 (4 x)}{7 x - 6}}{\frac{4 (7 x) - 4 (7 x - 6)}{7 x - 6}}$

Factor $4$ out of $- 4 (7 x - 6)$ .

$f^{- 1} (\frac{4 x}{7 x - 6}) = \frac{\frac{6 (4 x)}{7 x - 6}}{\frac{4 (7 x) + 4 (- (7 x - 6))}{7 x - 6}}$

Factor $4$ out of $4 (7 x) + 4 (- (7 x - 6))$ .

$f^{- 1} (\frac{4 x}{7 x - 6}) = \frac{\frac{6 (4 x)}{7 x - 6}}{\frac{4 (7 x - (7 x - 6))}{7 x - 6}}$

Apply the distributive property.

$f^{- 1} (\frac{4 x}{7 x - 6}) = \frac{\frac{6 (4 x)}{7 x - 6}}{\frac{4 (7 x - (7 x) + 6)}{7 x - 6}}$

Multiply $7$ by $- 1$ .

$f^{- 1} (\frac{4 x}{7 x - 6}) = \frac{\frac{6 (4 x)}{7 x - 6}}{\frac{4 (7 x - 7 x + 6)}{7 x - 6}}$

Multiply $- 1$ by $- 6$ .

$f^{- 1} (\frac{4 x}{7 x - 6}) = \frac{\frac{6 (4 x)}{7 x - 6}}{\frac{4 (7 x - 7 x + 6)}{7 x - 6}}$

Subtract $7 x$ from $7 x$ .

$f^{- 1} (\frac{4 x}{7 x - 6}) = \frac{\frac{6 (4 x)}{7 x - 6}}{\frac{4 (0 + 6)}{7 x - 6}}$

Add $0$ and $6$ .

$f^{- 1} (\frac{4 x}{7 x - 6}) = \frac{\frac{6 (4 x)}{7 x - 6}}{\frac{4 \cdot 6}{7 x - 6}}$

Multiply $4$ by $6$ .

$f^{- 1} (\frac{4 x}{7 x - 6}) = \frac{\frac{6 (4 x)}{7 x - 6}}{\frac{24}{7 x - 6}}$

Multiply $6$ by $4$ .

$f^{- 1} (\frac{4 x}{7 x - 6}) = \frac{\frac{24 x}{7 x - 6}}{\frac{24}{7 x - 6}}$

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{4 x}{7 x - 6}) = \frac{24 x}{7 x - 6} \cdot \frac{7 x - 6}{24}$

Cancel the common factor of $24$ .

Factor $24$ out of $24 x$ .

$f^{- 1} (\frac{4 x}{7 x - 6}) = \frac{24 (x)}{7 x - 6} \cdot \frac{7 x - 6}{24}$

Cancel the common factor.

$f^{- 1} (\frac{4 x}{7 x - 6}) = \frac{24 x}{7 x - 6} \cdot \frac{7 x - 6}{24}$

Rewrite the expression.

$f^{- 1} (\frac{4 x}{7 x - 6}) = \frac{x}{7 x - 6} \cdot (7 x - 6)$

Cancel the common factor of $7 x - 6$ .

Cancel the common factor.

$f^{- 1} (\frac{4 x}{7 x - 6}) = \frac{x}{7 x - 6} \cdot (7 x - 6)$

Rewrite the expression.

$f^{- 1} (\frac{4 x}{7 x - 6}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{6 x}{7 x - 4})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{6 x}{7 x - 4}) = \frac{4 (\frac{6 x}{7 x - 4})}{7 (\frac{6 x}{7 x - 4}) - 6}$

Combine $4$ and $\frac{6 x}{7 x - 4}$ .

$f (\frac{6 x}{7 x - 4}) = \frac{\frac{4 (6 x)}{7 x - 4}}{7 (\frac{6 x}{7 x - 4}) - 6}$

Simplify the denominator.

Multiply $7 \frac{6 x}{7 x - 4}$ .

Combine $7$ and $\frac{6 x}{7 x - 4}$ .

$f (\frac{6 x}{7 x - 4}) = \frac{\frac{4 (6 x)}{7 x - 4}}{\frac{7 (6 x)}{7 x - 4} - 6}$

Multiply $6$ by $7$ .

$f (\frac{6 x}{7 x - 4}) = \frac{\frac{4 (6 x)}{7 x - 4}}{\frac{42 x}{7 x - 4} - 6}$

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{7 x - 4}{7 x - 4}$ .

$f (\frac{6 x}{7 x - 4}) = \frac{\frac{4 (6 x)}{7 x - 4}}{\frac{42 x}{7 x - 4} - 6 \cdot \frac{7 x - 4}{7 x - 4}}$

Combine $- 6$ and $\frac{7 x - 4}{7 x - 4}$ .

$f (\frac{6 x}{7 x - 4}) = \frac{\frac{4 (6 x)}{7 x - 4}}{\frac{42 x}{7 x - 4} + \frac{- 6 (7 x - 4)}{7 x - 4}}$

Combine the numerators over the common denominator.

$f (\frac{6 x}{7 x - 4}) = \frac{\frac{4 (6 x)}{7 x - 4}}{\frac{42 x - 6 (7 x - 4)}{7 x - 4}}$

Rewrite $\frac{42 x - 6 (7 x - 4)}{7 x - 4}$ in a factored form.

Factor $6$ out of $42 x - 6 (7 x - 4)$ .

Factor $6$ out of $42 x$ .

$f (\frac{6 x}{7 x - 4}) = \frac{\frac{4 (6 x)}{7 x - 4}}{\frac{6 (7 x) - 6 (7 x - 4)}{7 x - 4}}$

Factor $6$ out of $- 6 (7 x - 4)$ .

$f (\frac{6 x}{7 x - 4}) = \frac{\frac{4 (6 x)}{7 x - 4}}{\frac{6 (7 x) + 6 (- (7 x - 4))}{7 x - 4}}$

Factor $6$ out of $6 (7 x) + 6 (- (7 x - 4))$ .

$f (\frac{6 x}{7 x - 4}) = \frac{\frac{4 (6 x)}{7 x - 4}}{\frac{6 (7 x - (7 x - 4))}{7 x - 4}}$

Apply the distributive property.

$f (\frac{6 x}{7 x - 4}) = \frac{\frac{4 (6 x)}{7 x - 4}}{\frac{6 (7 x - (7 x) + 4)}{7 x - 4}}$

Multiply $7$ by $- 1$ .

$f (\frac{6 x}{7 x - 4}) = \frac{\frac{4 (6 x)}{7 x - 4}}{\frac{6 (7 x - 7 x + 4)}{7 x - 4}}$

Multiply $- 1$ by $- 4$ .

$f (\frac{6 x}{7 x - 4}) = \frac{\frac{4 (6 x)}{7 x - 4}}{\frac{6 (7 x - 7 x + 4)}{7 x - 4}}$

Subtract $7 x$ from $7 x$ .

$f (\frac{6 x}{7 x - 4}) = \frac{\frac{4 (6 x)}{7 x - 4}}{\frac{6 (0 + 4)}{7 x - 4}}$

Add $0$ and $4$ .

$f (\frac{6 x}{7 x - 4}) = \frac{\frac{4 (6 x)}{7 x - 4}}{\frac{6 \cdot 4}{7 x - 4}}$

Multiply $6$ by $4$ .

$f (\frac{6 x}{7 x - 4}) = \frac{\frac{4 (6 x)}{7 x - 4}}{\frac{24}{7 x - 4}}$

Multiply $4$ by $6$ .

$f (\frac{6 x}{7 x - 4}) = \frac{\frac{24 x}{7 x - 4}}{\frac{24}{7 x - 4}}$

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{6 x}{7 x - 4}) = \frac{24 x}{7 x - 4} \cdot \frac{7 x - 4}{24}$

Cancel the common factor of $24$ .

Factor $24$ out of $24 x$ .

$f (\frac{6 x}{7 x - 4}) = \frac{24 (x)}{7 x - 4} \cdot \frac{7 x - 4}{24}$

Cancel the common factor.

$f (\frac{6 x}{7 x - 4}) = \frac{24 x}{7 x - 4} \cdot \frac{7 x - 4}{24}$

Rewrite the expression.

$f (\frac{6 x}{7 x - 4}) = \frac{x}{7 x - 4} \cdot (7 x - 4)$

Cancel the common factor of $7 x - 4$ .

Cancel the common factor.

$f (\frac{6 x}{7 x - 4}) = \frac{x}{7 x - 4} \cdot (7 x - 4)$

Rewrite the expression.

$f (\frac{6 x}{7 x - 4}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{6 x}{7 x - 4}$ is the inverse of $f (x) = \frac{4 x}{7 x - 6}$ .

$f^{- 1} (x) = \frac{6 x}{7 x - 4}$

Find the Inverse (4x)/(7x-6)

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