Find the Inverse (2x)/(7x-3)

$\frac{2 x}{7 x - 3}$

Step 1

Interchange the variables.

$x = \frac{2 y}{7 y - 3}$

Step 2

Solve for $y$ .

Multiply the equation by $7 y - 3$ .

$(7 y - 3) (x) = (\frac{2 y}{7 y - 3}) (7 y - 3)$

Simplify the left side.

Apply the distributive property.

$7 y x - 3 x = (\frac{2 y}{7 y - 3}) (7 y - 3)$

Simplify the right side.

Cancel the common factor of $7 y - 3$ .

Cancel the common factor.

$7 y x - 3 x = \frac{2 y}{7 y - 3} (7 y - 3)$

Rewrite the expression.

$7 y x - 3 x = 2 y$

Solve for $y$ .

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

$7 y x - 3 x - 2 y = 0$

Add $3 x$ to both sides of the equation.

$7 y x - 2 y = 3 x$

Factor $y$ out of $7 y x - 2 y$ .

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

$y (7 x) - 2 y = 3 x$

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

$y (7 x) + y \cdot - 2 = 3 x$

Factor $y$ out of $y (7 x) + y \cdot - 2$ .

$y (7 x - 2) = 3 x$

Divide each term in $y (7 x - 2) = 3 x$ by $7 x - 2$ and simplify.

Divide each term in $y (7 x - 2) = 3 x$ by $7 x - 2$ .

$\frac{y (7 x - 2)}{7 x - 2} = \frac{3 x}{7 x - 2}$

Simplify the left side.

Cancel the common factor of $7 x - 2$ .

Cancel the common factor.

$\frac{y (7 x - 2)}{7 x - 2} = \frac{3 x}{7 x - 2}$

Divide $y$ by $1$ .

$y = \frac{3 x}{7 x - 2}$

Step 3

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

$f^{- 1} (x) = \frac{3 x}{7 x - 2}$

Step 4

Verify if $f^{- 1} (x) = \frac{3 x}{7 x - 2}$ is the inverse of $f (x) = \frac{2 x}{7 x - 3}$ .

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 $f^{- 1} (\frac{2 x}{7 x - 3})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{2 x}{7 x - 3}) = \frac{3 (\frac{2 x}{7 x - 3})}{7 (\frac{2 x}{7 x - 3}) - 2}$

Combine $3$ and $\frac{2 x}{7 x - 3}$ .

$f^{- 1} (\frac{2 x}{7 x - 3}) = \frac{\frac{3 (2 x)}{7 x - 3}}{7 (\frac{2 x}{7 x - 3}) - 2}$

Simplify the denominator.

Multiply $7 \frac{2 x}{7 x - 3}$ .

Combine $7$ and $\frac{2 x}{7 x - 3}$ .

$f^{- 1} (\frac{2 x}{7 x - 3}) = \frac{\frac{3 (2 x)}{7 x - 3}}{\frac{7 (2 x)}{7 x - 3} - 2}$

Multiply $2$ by $7$ .

$f^{- 1} (\frac{2 x}{7 x - 3}) = \frac{\frac{3 (2 x)}{7 x - 3}}{\frac{14 x}{7 x - 3} - 2}$

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{7 x - 3}{7 x - 3}$ .

$f^{- 1} (\frac{2 x}{7 x - 3}) = \frac{\frac{3 (2 x)}{7 x - 3}}{\frac{14 x}{7 x - 3} - 2 \cdot \frac{7 x - 3}{7 x - 3}}$

Combine $- 2$ and $\frac{7 x - 3}{7 x - 3}$ .

$f^{- 1} (\frac{2 x}{7 x - 3}) = \frac{\frac{3 (2 x)}{7 x - 3}}{\frac{14 x}{7 x - 3} + \frac{- 2 (7 x - 3)}{7 x - 3}}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{2 x}{7 x - 3}) = \frac{\frac{3 (2 x)}{7 x - 3}}{\frac{14 x - 2 (7 x - 3)}{7 x - 3}}$

Rewrite $\frac{14 x - 2 (7 x - 3)}{7 x - 3}$ in a factored form.

Factor $2$ out of $14 x - 2 (7 x - 3)$ .

Factor $2$ out of $14 x$ .

$f^{- 1} (\frac{2 x}{7 x - 3}) = \frac{\frac{3 (2 x)}{7 x - 3}}{\frac{2 (7 x) - 2 (7 x - 3)}{7 x - 3}}$

Factor $2$ out of $- 2 (7 x - 3)$ .

$f^{- 1} (\frac{2 x}{7 x - 3}) = \frac{\frac{3 (2 x)}{7 x - 3}}{\frac{2 (7 x) + 2 (- (7 x - 3))}{7 x - 3}}$

Factor $2$ out of $2 (7 x) + 2 (- (7 x - 3))$ .

$f^{- 1} (\frac{2 x}{7 x - 3}) = \frac{\frac{3 (2 x)}{7 x - 3}}{\frac{2 (7 x - (7 x - 3))}{7 x - 3}}$

Apply the distributive property.

$f^{- 1} (\frac{2 x}{7 x - 3}) = \frac{\frac{3 (2 x)}{7 x - 3}}{\frac{2 (7 x - (7 x) + 3)}{7 x - 3}}$

Multiply $7$ by $- 1$ .

$f^{- 1} (\frac{2 x}{7 x - 3}) = \frac{\frac{3 (2 x)}{7 x - 3}}{\frac{2 (7 x - 7 x + 3)}{7 x - 3}}$

Multiply $- 1$ by $- 3$ .

$f^{- 1} (\frac{2 x}{7 x - 3}) = \frac{\frac{3 (2 x)}{7 x - 3}}{\frac{2 (7 x - 7 x + 3)}{7 x - 3}}$

Subtract $7 x$ from $7 x$ .

$f^{- 1} (\frac{2 x}{7 x - 3}) = \frac{\frac{3 (2 x)}{7 x - 3}}{\frac{2 (0 + 3)}{7 x - 3}}$

Add $0$ and $3$ .

$f^{- 1} (\frac{2 x}{7 x - 3}) = \frac{\frac{3 (2 x)}{7 x - 3}}{\frac{2 \cdot 3}{7 x - 3}}$

Multiply $2$ by $3$ .

$f^{- 1} (\frac{2 x}{7 x - 3}) = \frac{\frac{3 (2 x)}{7 x - 3}}{\frac{6}{7 x - 3}}$

Multiply $3$ by $2$ .

$f^{- 1} (\frac{2 x}{7 x - 3}) = \frac{\frac{6 x}{7 x - 3}}{\frac{6}{7 x - 3}}$

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{2 x}{7 x - 3}) = \frac{6 x}{7 x - 3} \cdot \frac{7 x - 3}{6}$

Cancel the common factor of $6$ .

Factor $6$ out of $6 x$ .

$f^{- 1} (\frac{2 x}{7 x - 3}) = \frac{6 (x)}{7 x - 3} \cdot \frac{7 x - 3}{6}$

Cancel the common factor.

$f^{- 1} (\frac{2 x}{7 x - 3}) = \frac{6 x}{7 x - 3} \cdot \frac{7 x - 3}{6}$

Rewrite the expression.

$f^{- 1} (\frac{2 x}{7 x - 3}) = \frac{x}{7 x - 3} \cdot (7 x - 3)$

Cancel the common factor of $7 x - 3$ .

Cancel the common factor.

$f^{- 1} (\frac{2 x}{7 x - 3}) = \frac{x}{7 x - 3} \cdot (7 x - 3)$

Rewrite the expression.

$f^{- 1} (\frac{2 x}{7 x - 3}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{3 x}{7 x - 2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{3 x}{7 x - 2}) = \frac{2 (\frac{3 x}{7 x - 2})}{7 (\frac{3 x}{7 x - 2}) - 3}$

Combine $2$ and $\frac{3 x}{7 x - 2}$ .

$f (\frac{3 x}{7 x - 2}) = \frac{\frac{2 (3 x)}{7 x - 2}}{7 (\frac{3 x}{7 x - 2}) - 3}$

Simplify the denominator.

Multiply $7 \frac{3 x}{7 x - 2}$ .

Combine $7$ and $\frac{3 x}{7 x - 2}$ .

$f (\frac{3 x}{7 x - 2}) = \frac{\frac{2 (3 x)}{7 x - 2}}{\frac{7 (3 x)}{7 x - 2} - 3}$

Multiply $3$ by $7$ .

$f (\frac{3 x}{7 x - 2}) = \frac{\frac{2 (3 x)}{7 x - 2}}{\frac{21 x}{7 x - 2} - 3}$

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{7 x - 2}{7 x - 2}$ .

$f (\frac{3 x}{7 x - 2}) = \frac{\frac{2 (3 x)}{7 x - 2}}{\frac{21 x}{7 x - 2} - 3 \cdot \frac{7 x - 2}{7 x - 2}}$

Combine $- 3$ and $\frac{7 x - 2}{7 x - 2}$ .

$f (\frac{3 x}{7 x - 2}) = \frac{\frac{2 (3 x)}{7 x - 2}}{\frac{21 x}{7 x - 2} + \frac{- 3 (7 x - 2)}{7 x - 2}}$

Combine the numerators over the common denominator.

$f (\frac{3 x}{7 x - 2}) = \frac{\frac{2 (3 x)}{7 x - 2}}{\frac{21 x - 3 (7 x - 2)}{7 x - 2}}$

Rewrite $\frac{21 x - 3 (7 x - 2)}{7 x - 2}$ in a factored form.

Factor $3$ out of $21 x - 3 (7 x - 2)$ .

Factor $3$ out of $21 x$ .

$f (\frac{3 x}{7 x - 2}) = \frac{\frac{2 (3 x)}{7 x - 2}}{\frac{3 (7 x) - 3 (7 x - 2)}{7 x - 2}}$

Factor $3$ out of $- 3 (7 x - 2)$ .

$f (\frac{3 x}{7 x - 2}) = \frac{\frac{2 (3 x)}{7 x - 2}}{\frac{3 (7 x) + 3 (- (7 x - 2))}{7 x - 2}}$

Factor $3$ out of $3 (7 x) + 3 (- (7 x - 2))$ .

$f (\frac{3 x}{7 x - 2}) = \frac{\frac{2 (3 x)}{7 x - 2}}{\frac{3 (7 x - (7 x - 2))}{7 x - 2}}$

Apply the distributive property.

$f (\frac{3 x}{7 x - 2}) = \frac{\frac{2 (3 x)}{7 x - 2}}{\frac{3 (7 x - (7 x) + 2)}{7 x - 2}}$

Multiply $7$ by $- 1$ .

$f (\frac{3 x}{7 x - 2}) = \frac{\frac{2 (3 x)}{7 x - 2}}{\frac{3 (7 x - 7 x + 2)}{7 x - 2}}$

Multiply $- 1$ by $- 2$ .

$f (\frac{3 x}{7 x - 2}) = \frac{\frac{2 (3 x)}{7 x - 2}}{\frac{3 (7 x - 7 x + 2)}{7 x - 2}}$

Subtract $7 x$ from $7 x$ .

$f (\frac{3 x}{7 x - 2}) = \frac{\frac{2 (3 x)}{7 x - 2}}{\frac{3 (0 + 2)}{7 x - 2}}$

Add $0$ and $2$ .

$f (\frac{3 x}{7 x - 2}) = \frac{\frac{2 (3 x)}{7 x - 2}}{\frac{3 \cdot 2}{7 x - 2}}$

Multiply $3$ by $2$ .

$f (\frac{3 x}{7 x - 2}) = \frac{\frac{2 (3 x)}{7 x - 2}}{\frac{6}{7 x - 2}}$

Multiply $2$ by $3$ .

$f (\frac{3 x}{7 x - 2}) = \frac{\frac{6 x}{7 x - 2}}{\frac{6}{7 x - 2}}$

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{3 x}{7 x - 2}) = \frac{6 x}{7 x - 2} \cdot \frac{7 x - 2}{6}$

Cancel the common factor of $6$ .

Factor $6$ out of $6 x$ .

$f (\frac{3 x}{7 x - 2}) = \frac{6 (x)}{7 x - 2} \cdot \frac{7 x - 2}{6}$

Cancel the common factor.

$f (\frac{3 x}{7 x - 2}) = \frac{6 x}{7 x - 2} \cdot \frac{7 x - 2}{6}$

Rewrite the expression.

$f (\frac{3 x}{7 x - 2}) = \frac{x}{7 x - 2} \cdot (7 x - 2)$

Cancel the common factor of $7 x - 2$ .

Cancel the common factor.

$f (\frac{3 x}{7 x - 2}) = \frac{x}{7 x - 2} \cdot (7 x - 2)$

Rewrite the expression.

$f (\frac{3 x}{7 x - 2}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{3 x}{7 x - 2}$ is the inverse of $f (x) = \frac{2 x}{7 x - 3}$ .

$f^{- 1} (x) = \frac{3 x}{7 x - 2}$

Find the Inverse (2x)/(7x-3)

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