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

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

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

Multiply the equation by $2 y - 3$ .

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

Simplify the left side.

Apply the distributive property.

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

Simplify the right side.

Cancel the common factor of $2 y - 3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Solve for $y$ .

Subtract $3 y$ from both sides of the equation.

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

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

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

Factor $y$ out of $2 y x - 3 y$ .

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

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

Factor $y$ out of $- 3 y$ .

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

Factor $y$ out of $y (2 x) + y \cdot - 3$ .

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

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

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

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

Simplify the left side.

Cancel the common factor of $2 x - 3$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Combine the numerators over the common denominator.

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

Step 3

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

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

Step 4

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

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

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

Simplify the numerator.

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

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

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

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

Combine the numerators over the common denominator.

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

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

Apply the distributive property.

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

Multiply $2$ by $7$ .

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

Multiply $7$ by $- 3$ .

$f^{- 1} (\frac{3 x + 7}{2 x - 3}) = \frac{\frac{14 x - 21 + 3 (3 x + 7)}{2 x - 3}}{2 (\frac{3 x + 7}{2 x - 3}) - 3}$

Apply the distributive property.

$f^{- 1} (\frac{3 x + 7}{2 x - 3}) = \frac{\frac{14 x - 21 + 3 (3 x) + 3 \cdot 7}{2 x - 3}}{2 (\frac{3 x + 7}{2 x - 3}) - 3}$

Multiply $3$ by $3$ .

$f^{- 1} (\frac{3 x + 7}{2 x - 3}) = \frac{\frac{14 x - 21 + 9 x + 3 \cdot 7}{2 x - 3}}{2 (\frac{3 x + 7}{2 x - 3}) - 3}$

Multiply $3$ by $7$ .

$f^{- 1} (\frac{3 x + 7}{2 x - 3}) = \frac{\frac{14 x - 21 + 9 x + 21}{2 x - 3}}{2 (\frac{3 x + 7}{2 x - 3}) - 3}$

Add $14 x$ and $9 x$ .

$f^{- 1} (\frac{3 x + 7}{2 x - 3}) = \frac{\frac{23 x - 21 + 21}{2 x - 3}}{2 (\frac{3 x + 7}{2 x - 3}) - 3}$

Add $- 21$ and $21$ .

$f^{- 1} (\frac{3 x + 7}{2 x - 3}) = \frac{\frac{23 x + 0}{2 x - 3}}{2 (\frac{3 x + 7}{2 x - 3}) - 3}$

Add $23 x$ and $0$ .

$f^{- 1} (\frac{3 x + 7}{2 x - 3}) = \frac{\frac{23 x}{2 x - 3}}{2 (\frac{3 x + 7}{2 x - 3}) - 3}$

Simplify the denominator.

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

$f^{- 1} (\frac{3 x + 7}{2 x - 3}) = \frac{\frac{23 x}{2 x - 3}}{\frac{2 (3 x + 7)}{2 x - 3} - 3}$

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

$f^{- 1} (\frac{3 x + 7}{2 x - 3}) = \frac{\frac{23 x}{2 x - 3}}{\frac{2 (3 x + 7)}{2 x - 3} - 3 \cdot \frac{2 x - 3}{2 x - 3}}$

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

$f^{- 1} (\frac{3 x + 7}{2 x - 3}) = \frac{\frac{23 x}{2 x - 3}}{\frac{2 (3 x + 7)}{2 x - 3} + \frac{- 3 (2 x - 3)}{2 x - 3}}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{3 x + 7}{2 x - 3}) = \frac{\frac{23 x}{2 x - 3}}{\frac{2 (3 x + 7) - 3 (2 x - 3)}{2 x - 3}}$

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

Apply the distributive property.

$f^{- 1} (\frac{3 x + 7}{2 x - 3}) = \frac{\frac{23 x}{2 x - 3}}{\frac{2 (3 x) + 2 \cdot 7 - 3 (2 x - 3)}{2 x - 3}}$

Multiply $3$ by $2$ .

$f^{- 1} (\frac{3 x + 7}{2 x - 3}) = \frac{\frac{23 x}{2 x - 3}}{\frac{6 x + 2 \cdot 7 - 3 (2 x - 3)}{2 x - 3}}$

Multiply $2$ by $7$ .

$f^{- 1} (\frac{3 x + 7}{2 x - 3}) = \frac{\frac{23 x}{2 x - 3}}{\frac{6 x + 14 - 3 (2 x - 3)}{2 x - 3}}$

Apply the distributive property.

$f^{- 1} (\frac{3 x + 7}{2 x - 3}) = \frac{\frac{23 x}{2 x - 3}}{\frac{6 x + 14 - 3 (2 x) - 3 \cdot - 3}{2 x - 3}}$

Multiply $2$ by $- 3$ .

$f^{- 1} (\frac{3 x + 7}{2 x - 3}) = \frac{\frac{23 x}{2 x - 3}}{\frac{6 x + 14 - 6 x - 3 \cdot - 3}{2 x - 3}}$

Multiply $- 3$ by $- 3$ .

$f^{- 1} (\frac{3 x + 7}{2 x - 3}) = \frac{\frac{23 x}{2 x - 3}}{\frac{6 x + 14 - 6 x + 9}{2 x - 3}}$

Subtract $6 x$ from $6 x$ .

$f^{- 1} (\frac{3 x + 7}{2 x - 3}) = \frac{\frac{23 x}{2 x - 3}}{\frac{0 + 14 + 9}{2 x - 3}}$

Add $0$ and $14$ .

$f^{- 1} (\frac{3 x + 7}{2 x - 3}) = \frac{\frac{23 x}{2 x - 3}}{\frac{14 + 9}{2 x - 3}}$

Add $14$ and $9$ .

$f^{- 1} (\frac{3 x + 7}{2 x - 3}) = \frac{\frac{23 x}{2 x - 3}}{\frac{23}{2 x - 3}}$

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{3 x + 7}{2 x - 3}) = \frac{23 x}{2 x - 3} \cdot \frac{2 x - 3}{23}$

Cancel the common factor of $23$ .

Factor $23$ out of $23 x$ .

$f^{- 1} (\frac{3 x + 7}{2 x - 3}) = \frac{23 (x)}{2 x - 3} \cdot \frac{2 x - 3}{23}$

Cancel the common factor.

$f^{- 1} (\frac{3 x + 7}{2 x - 3}) = \frac{23 x}{2 x - 3} \cdot \frac{2 x - 3}{23}$

Rewrite the expression.

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

Cancel the common factor of $2 x - 3$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

Set up the composite result function.

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

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

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

Simplify the numerator.

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

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

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

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

Combine the numerators over the common denominator.

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

Reorder terms.

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

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

Apply the distributive property.

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

Multiply $2$ by $7$ .

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

Multiply $7$ by $- 3$ .

$f (\frac{7 + 3 x}{2 x - 3}) = \frac{\frac{14 x - 21 + 3 (7 + 3 x)}{2 x - 3}}{2 (\frac{7 + 3 x}{2 x - 3}) - 3}$

Apply the distributive property.

$f (\frac{7 + 3 x}{2 x - 3}) = \frac{\frac{14 x - 21 + 3 \cdot 7 + 3 (3 x)}{2 x - 3}}{2 (\frac{7 + 3 x}{2 x - 3}) - 3}$

Multiply $3$ by $7$ .

$f (\frac{7 + 3 x}{2 x - 3}) = \frac{\frac{14 x - 21 + 21 + 3 (3 x)}{2 x - 3}}{2 (\frac{7 + 3 x}{2 x - 3}) - 3}$

Multiply $3$ by $3$ .

$f (\frac{7 + 3 x}{2 x - 3}) = \frac{\frac{14 x - 21 + 21 + 9 x}{2 x - 3}}{2 (\frac{7 + 3 x}{2 x - 3}) - 3}$

Add $14 x$ and $9 x$ .

$f (\frac{7 + 3 x}{2 x - 3}) = \frac{\frac{23 x - 21 + 21}{2 x - 3}}{2 (\frac{7 + 3 x}{2 x - 3}) - 3}$

Add $- 21$ and $21$ .

$f (\frac{7 + 3 x}{2 x - 3}) = \frac{\frac{23 x + 0}{2 x - 3}}{2 (\frac{7 + 3 x}{2 x - 3}) - 3}$

Add $23 x$ and $0$ .

$f (\frac{7 + 3 x}{2 x - 3}) = \frac{\frac{23 x}{2 x - 3}}{2 (\frac{7 + 3 x}{2 x - 3}) - 3}$

Simplify the denominator.

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

$f (\frac{7 + 3 x}{2 x - 3}) = \frac{\frac{23 x}{2 x - 3}}{\frac{2 (7 + 3 x)}{2 x - 3} - 3}$

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

$f (\frac{7 + 3 x}{2 x - 3}) = \frac{\frac{23 x}{2 x - 3}}{\frac{2 (7 + 3 x)}{2 x - 3} - 3 \cdot \frac{2 x - 3}{2 x - 3}}$

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

$f (\frac{7 + 3 x}{2 x - 3}) = \frac{\frac{23 x}{2 x - 3}}{\frac{2 (7 + 3 x)}{2 x - 3} + \frac{- 3 (2 x - 3)}{2 x - 3}}$

Combine the numerators over the common denominator.

$f (\frac{7 + 3 x}{2 x - 3}) = \frac{\frac{23 x}{2 x - 3}}{\frac{2 (7 + 3 x) - 3 (2 x - 3)}{2 x - 3}}$

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

Apply the distributive property.

$f (\frac{7 + 3 x}{2 x - 3}) = \frac{\frac{23 x}{2 x - 3}}{\frac{2 \cdot 7 + 2 (3 x) - 3 (2 x - 3)}{2 x - 3}}$

Multiply $2$ by $7$ .

$f (\frac{7 + 3 x}{2 x - 3}) = \frac{\frac{23 x}{2 x - 3}}{\frac{14 + 2 (3 x) - 3 (2 x - 3)}{2 x - 3}}$

Multiply $3$ by $2$ .

$f (\frac{7 + 3 x}{2 x - 3}) = \frac{\frac{23 x}{2 x - 3}}{\frac{14 + 6 x - 3 (2 x - 3)}{2 x - 3}}$

Apply the distributive property.

$f (\frac{7 + 3 x}{2 x - 3}) = \frac{\frac{23 x}{2 x - 3}}{\frac{14 + 6 x - 3 (2 x) - 3 \cdot - 3}{2 x - 3}}$

Multiply $2$ by $- 3$ .

$f (\frac{7 + 3 x}{2 x - 3}) = \frac{\frac{23 x}{2 x - 3}}{\frac{14 + 6 x - 6 x - 3 \cdot - 3}{2 x - 3}}$

Multiply $- 3$ by $- 3$ .

$f (\frac{7 + 3 x}{2 x - 3}) = \frac{\frac{23 x}{2 x - 3}}{\frac{14 + 6 x - 6 x + 9}{2 x - 3}}$

Add $14$ and $9$ .

$f (\frac{7 + 3 x}{2 x - 3}) = \frac{\frac{23 x}{2 x - 3}}{\frac{6 x - 6 x + 23}{2 x - 3}}$

Subtract $6 x$ from $6 x$ .

$f (\frac{7 + 3 x}{2 x - 3}) = \frac{\frac{23 x}{2 x - 3}}{\frac{0 + 23}{2 x - 3}}$

Add $0$ and $23$ .

$f (\frac{7 + 3 x}{2 x - 3}) = \frac{\frac{23 x}{2 x - 3}}{\frac{23}{2 x - 3}}$

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{7 + 3 x}{2 x - 3}) = \frac{23 x}{2 x - 3} \cdot \frac{2 x - 3}{23}$

Cancel the common factor of $23$ .

Factor $23$ out of $23 x$ .

$f (\frac{7 + 3 x}{2 x - 3}) = \frac{23 (x)}{2 x - 3} \cdot \frac{2 x - 3}{23}$

Cancel the common factor.

$f (\frac{7 + 3 x}{2 x - 3}) = \frac{23 x}{2 x - 3} \cdot \frac{2 x - 3}{23}$

Rewrite the expression.

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

Cancel the common factor of $2 x - 3$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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