Find the Inverse f(x)=(7+3x)/(5x+6)

$f (x) = \frac{7 + 3 x}{5 x + 6}$

Step 1

Write $f (x) = \frac{7 + 3 x}{5 x + 6}$ as an equation.

$y = \frac{7 + 3 x}{5 x + 6}$

Step 2

Interchange the variables.

$x = \frac{7 + 3 y}{5 y + 6}$

Step 3

Solve for $y$ .

Rewrite the equation as $\frac{7 + 3 y}{5 y + 6} = x$ .

$\frac{7 + 3 y}{5 y + 6} = x$

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$5 y + 6, 1$

Remove parentheses.

$5 y + 6, 1$

The LCM of one and any expression is the expression.

$5 y + 6$

Multiply each term in $\frac{7 + 3 y}{5 y + 6} = x$ by $5 y + 6$ to eliminate the fractions.

Multiply each term in $\frac{7 + 3 y}{5 y + 6} = x$ by $5 y + 6$ .

$\frac{7 + 3 y}{5 y + 6} (5 y + 6) = x (5 y + 6)$

Simplify the left side.

Cancel the common factor of $5 y + 6$ .

Cancel the common factor.

$\frac{7 + 3 y}{5 y + 6} (5 y + 6) = x (5 y + 6)$

Rewrite the expression.

$7 + 3 y = x (5 y + 6)$

Simplify the right side.

Apply the distributive property.

$7 + 3 y = x (5 y) + x \cdot 6$

Reorder.

Rewrite using the commutative property of multiplication.

$7 + 3 y = 5 x y + x \cdot 6$

Move $6$ to the left of $x$ .

$7 + 3 y = 5 x y + 6 x$

Solve the equation.

Subtract $5 x y$ from both sides of the equation.

$7 + 3 y - 5 x y = 6 x$

Subtract $7$ from both sides of the equation.

$3 y - 5 x y = 6 x - 7$

Factor $y$ out of $3 y - 5 x y$ .

Factor $y$ out of $3 y$ .

$y \cdot 3 - 5 x y = 6 x - 7$

Factor $y$ out of $- 5 x y$ .

$y \cdot 3 + y (- 5 x) = 6 x - 7$

Factor $y$ out of $y \cdot 3 + y (- 5 x)$ .

$y (3 - 5 x) = 6 x - 7$

Divide each term in $y (3 - 5 x) = 6 x - 7$ by $3 - 5 x$ and simplify.

Divide each term in $y (3 - 5 x) = 6 x - 7$ by $3 - 5 x$ .

$\frac{y (3 - 5 x)}{3 - 5 x} = \frac{6 x}{3 - 5 x} + \frac{- 7}{3 - 5 x}$

Simplify the left side.

Cancel the common factor of $3 - 5 x$ .

Cancel the common factor.

$\frac{y (3 - 5 x)}{3 - 5 x} = \frac{6 x}{3 - 5 x} + \frac{- 7}{3 - 5 x}$

Divide $y$ by $1$ .

$y = \frac{6 x}{3 - 5 x} + \frac{- 7}{3 - 5 x}$

Simplify the right side.

Combine the numerators over the common denominator.

$y = \frac{6 x - 7}{3 - 5 x}$

Step 4

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

$f^{- 1} (x) = \frac{6 x - 7}{3 - 5 x}$

Step 5

Verify if $f^{- 1} (x) = \frac{6 x - 7}{3 - 5 x}$ is the inverse of $f (x) = \frac{7 + 3 x}{5 x + 6}$ .

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 $f^{- 1} (\frac{7 + 3 x}{5 x + 6})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{7 + 3 x}{5 x + 6}) = \frac{6 (\frac{7 + 3 x}{5 x + 6}) - 7}{3 - 5 \frac{7 + 3 x}{5 x + 6}}$

Simplify the numerator.

Combine $6$ and $\frac{7 + 3 x}{5 x + 6}$ .

$f^{- 1} (\frac{7 + 3 x}{5 x + 6}) = \frac{\frac{6 (7 + 3 x)}{5 x + 6} - 7}{3 - 5 \frac{7 + 3 x}{5 x + 6}}$

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{5 x + 6}{5 x + 6}$ .

$f^{- 1} (\frac{7 + 3 x}{5 x + 6}) = \frac{\frac{6 (7 + 3 x)}{5 x + 6} - 7 \cdot \frac{5 x + 6}{5 x + 6}}{3 - 5 \frac{7 + 3 x}{5 x + 6}}$

Combine $- 7$ and $\frac{5 x + 6}{5 x + 6}$ .

$f^{- 1} (\frac{7 + 3 x}{5 x + 6}) = \frac{\frac{6 (7 + 3 x)}{5 x + 6} + \frac{- 7 (5 x + 6)}{5 x + 6}}{3 - 5 \frac{7 + 3 x}{5 x + 6}}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{7 + 3 x}{5 x + 6}) = \frac{\frac{6 (7 + 3 x) - 7 (5 x + 6)}{5 x + 6}}{3 - 5 \frac{7 + 3 x}{5 x + 6}}$

Rewrite $\frac{6 (7 + 3 x) - 7 (5 x + 6)}{5 x + 6}$ in a factored form.

Apply the distributive property.

$f^{- 1} (\frac{7 + 3 x}{5 x + 6}) = \frac{\frac{6 \cdot 7 + 6 (3 x) - 7 (5 x + 6)}{5 x + 6}}{3 - 5 \frac{7 + 3 x}{5 x + 6}}$

Multiply $6$ by $7$ .

$f^{- 1} (\frac{7 + 3 x}{5 x + 6}) = \frac{\frac{42 + 6 (3 x) - 7 (5 x + 6)}{5 x + 6}}{3 - 5 \frac{7 + 3 x}{5 x + 6}}$

Multiply $3$ by $6$ .

$f^{- 1} (\frac{7 + 3 x}{5 x + 6}) = \frac{\frac{42 + 18 x - 7 (5 x + 6)}{5 x + 6}}{3 - 5 \frac{7 + 3 x}{5 x + 6}}$

Apply the distributive property.

$f^{- 1} (\frac{7 + 3 x}{5 x + 6}) = \frac{\frac{42 + 18 x - 7 (5 x) - 7 \cdot 6}{5 x + 6}}{3 - 5 \frac{7 + 3 x}{5 x + 6}}$

Multiply $5$ by $- 7$ .

$f^{- 1} (\frac{7 + 3 x}{5 x + 6}) = \frac{\frac{42 + 18 x - 35 x - 7 \cdot 6}{5 x + 6}}{3 - 5 \frac{7 + 3 x}{5 x + 6}}$

Multiply $- 7$ by $6$ .

$f^{- 1} (\frac{7 + 3 x}{5 x + 6}) = \frac{\frac{42 + 18 x - 35 x - 42}{5 x + 6}}{3 - 5 \frac{7 + 3 x}{5 x + 6}}$

Subtract $42$ from $42$ .

$f^{- 1} (\frac{7 + 3 x}{5 x + 6}) = \frac{\frac{18 x - 35 x + 0}{5 x + 6}}{3 - 5 \frac{7 + 3 x}{5 x + 6}}$

Add $18 x - 35 x$ and $0$ .

$f^{- 1} (\frac{7 + 3 x}{5 x + 6}) = \frac{\frac{18 x - 35 x}{5 x + 6}}{3 - 5 \frac{7 + 3 x}{5 x + 6}}$

Subtract $35 x$ from $18 x$ .

$f^{- 1} (\frac{7 + 3 x}{5 x + 6}) = \frac{\frac{- 17 x}{5 x + 6}}{3 - 5 \frac{7 + 3 x}{5 x + 6}}$

Move the negative in front of the fraction.

$f^{- 1} (\frac{7 + 3 x}{5 x + 6}) = \frac{- \frac{17 x}{5 x + 6}}{3 - 5 \frac{7 + 3 x}{5 x + 6}}$

Simplify the denominator.

Combine $- 5$ and $\frac{7 + 3 x}{5 x + 6}$ .

$f^{- 1} (\frac{7 + 3 x}{5 x + 6}) = \frac{- \frac{17 x}{5 x + 6}}{3 + \frac{- 5 (7 + 3 x)}{5 x + 6}}$

Move the negative in front of the fraction.

$f^{- 1} (\frac{7 + 3 x}{5 x + 6}) = \frac{- \frac{17 x}{5 x + 6}}{3 - \frac{(5) (7 + 3 x)}{5 x + 6}}$

To write $3$ as a fraction with a common denominator, multiply by $\frac{5 x + 6}{5 x + 6}$ .

$f^{- 1} (\frac{7 + 3 x}{5 x + 6}) = \frac{- \frac{17 x}{5 x + 6}}{\frac{3 (5 x + 6)}{5 x + 6} - \frac{5 (7 + 3 x)}{5 x + 6}}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{7 + 3 x}{5 x + 6}) = \frac{- \frac{17 x}{5 x + 6}}{\frac{3 (5 x + 6) - 5 (7 + 3 x)}{5 x + 6}}$

Rewrite $\frac{3 (5 x + 6) - 5 (7 + 3 x)}{5 x + 6}$ in a factored form.

Apply the distributive property.

$f^{- 1} (\frac{7 + 3 x}{5 x + 6}) = \frac{- \frac{17 x}{5 x + 6}}{\frac{3 (5 x) + 3 \cdot 6 - 5 (7 + 3 x)}{5 x + 6}}$

Multiply $5$ by $3$ .

$f^{- 1} (\frac{7 + 3 x}{5 x + 6}) = \frac{- \frac{17 x}{5 x + 6}}{\frac{15 x + 3 \cdot 6 - 5 (7 + 3 x)}{5 x + 6}}$

Multiply $3$ by $6$ .

$f^{- 1} (\frac{7 + 3 x}{5 x + 6}) = \frac{- \frac{17 x}{5 x + 6}}{\frac{15 x + 18 - 5 (7 + 3 x)}{5 x + 6}}$

Apply the distributive property.

$f^{- 1} (\frac{7 + 3 x}{5 x + 6}) = \frac{- \frac{17 x}{5 x + 6}}{\frac{15 x + 18 - 5 \cdot 7 - 5 (3 x)}{5 x + 6}}$

Multiply $- 5$ by $7$ .

$f^{- 1} (\frac{7 + 3 x}{5 x + 6}) = \frac{- \frac{17 x}{5 x + 6}}{\frac{15 x + 18 - 35 - 5 (3 x)}{5 x + 6}}$

Multiply $3$ by $- 5$ .

$f^{- 1} (\frac{7 + 3 x}{5 x + 6}) = \frac{- \frac{17 x}{5 x + 6}}{\frac{15 x + 18 - 35 - 15 x}{5 x + 6}}$

Subtract $15 x$ from $15 x$ .

$f^{- 1} (\frac{7 + 3 x}{5 x + 6}) = \frac{- \frac{17 x}{5 x + 6}}{\frac{0 + 18 - 35}{5 x + 6}}$

Add $0$ and $18$ .

$f^{- 1} (\frac{7 + 3 x}{5 x + 6}) = \frac{- \frac{17 x}{5 x + 6}}{\frac{18 - 35}{5 x + 6}}$

Subtract $35$ from $18$ .

$f^{- 1} (\frac{7 + 3 x}{5 x + 6}) = \frac{- \frac{17 x}{5 x + 6}}{\frac{- 17}{5 x + 6}}$

Move the negative in front of the fraction.

$f^{- 1} (\frac{7 + 3 x}{5 x + 6}) = \frac{- \frac{17 x}{5 x + 6}}{- \frac{17}{5 x + 6}}$

Dividing two negative values results in a positive value.

$f^{- 1} (\frac{7 + 3 x}{5 x + 6}) = \frac{\frac{17 x}{5 x + 6}}{\frac{17}{5 x + 6}}$

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{7 + 3 x}{5 x + 6}) = \frac{17 x}{5 x + 6} \cdot \frac{5 x + 6}{17}$

Cancel the common factor of $17$ .

Factor $17$ out of $17 x$ .

$f^{- 1} (\frac{7 + 3 x}{5 x + 6}) = \frac{17 (x)}{5 x + 6} \cdot \frac{5 x + 6}{17}$

Cancel the common factor.

$f^{- 1} (\frac{7 + 3 x}{5 x + 6}) = \frac{17 x}{5 x + 6} \cdot \frac{5 x + 6}{17}$

Rewrite the expression.

$f^{- 1} (\frac{7 + 3 x}{5 x + 6}) = \frac{x}{5 x + 6} \cdot (5 x + 6)$

Cancel the common factor of $5 x + 6$ .

Cancel the common factor.

$f^{- 1} (\frac{7 + 3 x}{5 x + 6}) = \frac{x}{5 x + 6} \cdot (5 x + 6)$

Rewrite the expression.

$f^{- 1} (\frac{7 + 3 x}{5 x + 6}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{6 x - 7}{3 - 5 x})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{6 x - 7}{3 - 5 x}) = \frac{7 + 3 (\frac{6 x - 7}{3 - 5 x})}{5 (\frac{6 x - 7}{3 - 5 x}) + 6}$

Simplify the numerator.

Combine $3$ and $\frac{6 x - 7}{3 - 5 x}$ .

$f (\frac{6 x - 7}{3 - 5 x}) = \frac{7 + \frac{3 (6 x - 7)}{3 - 5 x}}{5 (\frac{6 x - 7}{3 - 5 x}) + 6}$

To write $7$ as a fraction with a common denominator, multiply by $\frac{3 - 5 x}{3 - 5 x}$ .

$f (\frac{6 x - 7}{3 - 5 x}) = \frac{\frac{7 (3 - 5 x)}{3 - 5 x} + \frac{3 (6 x - 7)}{3 - 5 x}}{5 (\frac{6 x - 7}{3 - 5 x}) + 6}$

Combine the numerators over the common denominator.

$f (\frac{6 x - 7}{3 - 5 x}) = \frac{\frac{7 (3 - 5 x) + 3 (6 x - 7)}{3 - 5 x}}{5 (\frac{6 x - 7}{3 - 5 x}) + 6}$

Reorder terms.

$f (\frac{6 x - 7}{3 - 5 x}) = \frac{\frac{7 (3 - 5 x) + 3 (6 x - 7)}{- 5 x + 3}}{5 (\frac{6 x - 7}{3 - 5 x}) + 6}$

Rewrite $\frac{7 (3 - 5 x) + 3 (6 x - 7)}{- 5 x + 3}$ in a factored form.

Apply the distributive property.

$f (\frac{6 x - 7}{3 - 5 x}) = \frac{\frac{7 \cdot 3 + 7 (- 5 x) + 3 (6 x - 7)}{- 5 x + 3}}{5 (\frac{6 x - 7}{3 - 5 x}) + 6}$

Multiply $7$ by $3$ .

$f (\frac{6 x - 7}{3 - 5 x}) = \frac{\frac{21 + 7 (- 5 x) + 3 (6 x - 7)}{- 5 x + 3}}{5 (\frac{6 x - 7}{3 - 5 x}) + 6}$

Multiply $- 5$ by $7$ .

$f (\frac{6 x - 7}{3 - 5 x}) = \frac{\frac{21 - 35 x + 3 (6 x - 7)}{- 5 x + 3}}{5 (\frac{6 x - 7}{3 - 5 x}) + 6}$

Apply the distributive property.

$f (\frac{6 x - 7}{3 - 5 x}) = \frac{\frac{21 - 35 x + 3 (6 x) + 3 \cdot - 7}{- 5 x + 3}}{5 (\frac{6 x - 7}{3 - 5 x}) + 6}$

Multiply $6$ by $3$ .

$f (\frac{6 x - 7}{3 - 5 x}) = \frac{\frac{21 - 35 x + 18 x + 3 \cdot - 7}{- 5 x + 3}}{5 (\frac{6 x - 7}{3 - 5 x}) + 6}$

Multiply $3$ by $- 7$ .

$f (\frac{6 x - 7}{3 - 5 x}) = \frac{\frac{21 - 35 x + 18 x - 21}{- 5 x + 3}}{5 (\frac{6 x - 7}{3 - 5 x}) + 6}$

Subtract $21$ from $21$ .

$f (\frac{6 x - 7}{3 - 5 x}) = \frac{\frac{- 35 x + 18 x + 0}{- 5 x + 3}}{5 (\frac{6 x - 7}{3 - 5 x}) + 6}$

Add $- 35 x + 18 x$ and $0$ .

$f (\frac{6 x - 7}{3 - 5 x}) = \frac{\frac{- 35 x + 18 x}{- 5 x + 3}}{5 (\frac{6 x - 7}{3 - 5 x}) + 6}$

Add $- 35 x$ and $18 x$ .

$f (\frac{6 x - 7}{3 - 5 x}) = \frac{\frac{- 17 x}{- 5 x + 3}}{5 (\frac{6 x - 7}{3 - 5 x}) + 6}$

Move the negative in front of the fraction.

$f (\frac{6 x - 7}{3 - 5 x}) = \frac{- \frac{17 x}{- 5 x + 3}}{5 (\frac{6 x - 7}{3 - 5 x}) + 6}$

Simplify the denominator.

Combine $5$ and $\frac{6 x - 7}{3 - 5 x}$ .

$f (\frac{6 x - 7}{3 - 5 x}) = \frac{- \frac{17 x}{- 5 x + 3}}{\frac{5 (6 x - 7)}{3 - 5 x} + 6}$

To write $6$ as a fraction with a common denominator, multiply by $\frac{3 - 5 x}{3 - 5 x}$ .

$f (\frac{6 x - 7}{3 - 5 x}) = \frac{- \frac{17 x}{- 5 x + 3}}{\frac{5 (6 x - 7)}{3 - 5 x} + \frac{6 (3 - 5 x)}{3 - 5 x}}$

Combine the numerators over the common denominator.

$f (\frac{6 x - 7}{3 - 5 x}) = \frac{- \frac{17 x}{- 5 x + 3}}{\frac{5 (6 x - 7) + 6 (3 - 5 x)}{3 - 5 x}}$

Rewrite $\frac{5 (6 x - 7) + 6 (3 - 5 x)}{3 - 5 x}$ in a factored form.

Apply the distributive property.

$f (\frac{6 x - 7}{3 - 5 x}) = \frac{- \frac{17 x}{- 5 x + 3}}{\frac{5 (6 x) + 5 \cdot - 7 + 6 (3 - 5 x)}{3 - 5 x}}$

Multiply $6$ by $5$ .

$f (\frac{6 x - 7}{3 - 5 x}) = \frac{- \frac{17 x}{- 5 x + 3}}{\frac{30 x + 5 \cdot - 7 + 6 (3 - 5 x)}{3 - 5 x}}$

Multiply $5$ by $- 7$ .

$f (\frac{6 x - 7}{3 - 5 x}) = \frac{- \frac{17 x}{- 5 x + 3}}{\frac{30 x - 35 + 6 (3 - 5 x)}{3 - 5 x}}$

Apply the distributive property.

$f (\frac{6 x - 7}{3 - 5 x}) = \frac{- \frac{17 x}{- 5 x + 3}}{\frac{30 x - 35 + 6 \cdot 3 + 6 (- 5 x)}{3 - 5 x}}$

Multiply $6$ by $3$ .

$f (\frac{6 x - 7}{3 - 5 x}) = \frac{- \frac{17 x}{- 5 x + 3}}{\frac{30 x - 35 + 18 + 6 (- 5 x)}{3 - 5 x}}$

Multiply $- 5$ by $6$ .

$f (\frac{6 x - 7}{3 - 5 x}) = \frac{- \frac{17 x}{- 5 x + 3}}{\frac{30 x - 35 + 18 - 30 x}{3 - 5 x}}$

Subtract $30 x$ from $30 x$ .

$f (\frac{6 x - 7}{3 - 5 x}) = \frac{- \frac{17 x}{- 5 x + 3}}{\frac{0 - 35 + 18}{3 - 5 x}}$

Subtract $35$ from $0$ .

$f (\frac{6 x - 7}{3 - 5 x}) = \frac{- \frac{17 x}{- 5 x + 3}}{\frac{- 35 + 18}{3 - 5 x}}$

Add $- 35$ and $18$ .

$f (\frac{6 x - 7}{3 - 5 x}) = \frac{- \frac{17 x}{- 5 x + 3}}{\frac{- 17}{3 - 5 x}}$

Move the negative in front of the fraction.

$f (\frac{6 x - 7}{3 - 5 x}) = \frac{- \frac{17 x}{- 5 x + 3}}{- \frac{17}{3 - 5 x}}$

Dividing two negative values results in a positive value.

$f (\frac{6 x - 7}{3 - 5 x}) = \frac{\frac{17 x}{- 5 x + 3}}{\frac{17}{3 - 5 x}}$

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{6 x - 7}{3 - 5 x}) = \frac{17 x}{- 5 x + 3} \cdot \frac{3 - 5 x}{17}$

Cancel the common factor of $17$ .

Factor $17$ out of $17 x$ .

$f (\frac{6 x - 7}{3 - 5 x}) = \frac{17 (x)}{- 5 x + 3} \cdot \frac{3 - 5 x}{17}$

Cancel the common factor.

$f (\frac{6 x - 7}{3 - 5 x}) = \frac{17 x}{- 5 x + 3} \cdot \frac{3 - 5 x}{17}$

Rewrite the expression.

$f (\frac{6 x - 7}{3 - 5 x}) = \frac{x}{- 5 x + 3} \cdot (3 - 5 x)$

Multiply $\frac{x}{- 5 x + 3}$ by $3 - 5 x$ .

$f (\frac{6 x - 7}{3 - 5 x}) = \frac{x (3 - 5 x)}{- 5 x + 3}$

Cancel the common factor of $3 - 5 x$ and $- 5 x + 3$ .

Reorder terms.

$f (\frac{6 x - 7}{3 - 5 x}) = \frac{x (- 5 x + 3)}{- 5 x + 3}$

Cancel the common factor.

$f (\frac{6 x - 7}{3 - 5 x}) = \frac{x (- 5 x + 3)}{- 5 x + 3}$

Divide $x$ by $1$ .

$f (\frac{6 x - 7}{3 - 5 x}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{6 x - 7}{3 - 5 x}$ is the inverse of $f (x) = \frac{7 + 3 x}{5 x + 6}$ .

$f^{- 1} (x) = \frac{6 x - 7}{3 - 5 x}$

Find the Inverse f(x)=(7+3x)/(5x+6)

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