Find the Inverse (3x-9)/(-9x+7)

$\frac{3 x - 9}{- 9 x + 7}$

Step 1

Interchange the variables.

$x = \frac{3 y - 9}{- 9 y + 7}$

Step 2

Solve for $y$ .

Rewrite the equation as $\frac{3 y - 9}{- 9 y + 7} = x$ .

$\frac{3 y - 9}{- 9 y + 7} = x$

Factor $3$ out of $3 y - 9$ .

Factor $3$ out of $3 y$ .

$\frac{3 (y) - 9}{- 9 y + 7} = x$

Factor $3$ out of $- 9$ .

$\frac{3 y + 3 \cdot - 3}{- 9 y + 7} = x$

Factor $3$ out of $3 y + 3 \cdot - 3$ .

$\frac{3 (y - 3)}{- 9 y + 7} = x$

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.

$- 9 y + 7, 1$

Remove parentheses.

$- 9 y + 7, 1$

The LCM of one and any expression is the expression.

$- 9 y + 7$

Multiply each term in $\frac{3 (y - 3)}{- 9 y + 7} = x$ by $- 9 y + 7$ to eliminate the fractions.

Multiply each term in $\frac{3 (y - 3)}{- 9 y + 7} = x$ by $- 9 y + 7$ .

$\frac{3 (y - 3)}{- 9 y + 7} (- 9 y + 7) = x (- 9 y + 7)$

Simplify the left side.

Multiply $\frac{3 (y - 3)}{- 9 y + 7}$ by $- 9 y + 7$ .

$\frac{3 (y - 3) (- 9 y + 7)}{- 9 y + 7} = x (- 9 y + 7)$

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

Cancel the common factor.

$\frac{3 (y - 3) (- 9 y + 7)}{- 9 y + 7} = x (- 9 y + 7)$

Divide $3 (y - 3)$ by $1$ .

$3 (y - 3) = x (- 9 y + 7)$

Apply the distributive property.

$3 y + 3 \cdot - 3 = x (- 9 y + 7)$

Multiply $3$ by $- 3$ .

$3 y - 9 = x (- 9 y + 7)$

Simplify the right side.

Apply the distributive property.

$3 y - 9 = x (- 9 y) + x \cdot 7$

Reorder.

Rewrite using the commutative property of multiplication.

$3 y - 9 = - 9 x y + x \cdot 7$

Move $7$ to the left of $x$ .

$3 y - 9 = - 9 x y + 7 x$

Solve the equation.

Add $9 x y$ to both sides of the equation.

$3 y - 9 + 9 x y = 7 x$

Add $9$ to both sides of the equation.

$3 y + 9 x y = 7 x + 9$

Factor $3 y$ out of $3 y + 9 x y$ .

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

$3 y (1) + 9 x y = 7 x + 9$

Factor $3 y$ out of $9 x y$ .

$3 y (1) + 3 y (3 x) = 7 x + 9$

Factor $3 y$ out of $3 y (1) + 3 y (3 x)$ .

$3 y (1 + 3 x) = 7 x + 9$

Divide each term in $3 y (1 + 3 x) = 7 x + 9$ by $3 (1 + 3 x)$ and simplify.

Divide each term in $3 y (1 + 3 x) = 7 x + 9$ by $3 (1 + 3 x)$ .

$\frac{3 y (1 + 3 x)}{3 (1 + 3 x)} = \frac{7 x}{3 (1 + 3 x)} + \frac{9}{3 (1 + 3 x)}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 y (1 + 3 x)}{3 (1 + 3 x)} = \frac{7 x}{3 (1 + 3 x)} + \frac{9}{3 (1 + 3 x)}$

Rewrite the expression.

$\frac{y (1 + 3 x)}{1 + 3 x} = \frac{7 x}{3 (1 + 3 x)} + \frac{9}{3 (1 + 3 x)}$

Cancel the common factor of $1 + 3 x$ .

Cancel the common factor.

$\frac{y (1 + 3 x)}{1 + 3 x} = \frac{7 x}{3 (1 + 3 x)} + \frac{9}{3 (1 + 3 x)}$

Divide $y$ by $1$ .

$y = \frac{7 x}{3 (1 + 3 x)} + \frac{9}{3 (1 + 3 x)}$

Simplify the right side.

Cancel the common factor of $9$ and $3$ .

Factor $3$ out of $9$ .

$y = \frac{7 x}{3 (1 + 3 x)} + \frac{3 \cdot 3}{3 (1 + 3 x)}$

Cancel the common factors.

Cancel the common factor.

$y = \frac{7 x}{3 (1 + 3 x)} + \frac{3 \cdot 3}{3 (1 + 3 x)}$

Rewrite the expression.

$y = \frac{7 x}{3 (1 + 3 x)} + \frac{3}{1 + 3 x}$

To write $\frac{3}{1 + 3 x}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \frac{7 x}{3 (1 + 3 x)} + \frac{3}{1 + 3 x} \cdot \frac{3}{3}$

Write each expression with a common denominator of $3 (1 + 3 x)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3}{1 + 3 x}$ by $\frac{3}{3}$ .

$y = \frac{7 x}{3 (1 + 3 x)} + \frac{3 \cdot 3}{(1 + 3 x) \cdot 3}$

Reorder the factors of $(1 + 3 x) \cdot 3$ .

$y = \frac{7 x}{3 (1 + 3 x)} + \frac{3 \cdot 3}{3 (1 + 3 x)}$

Combine the numerators over the common denominator.

$y = \frac{7 x + 3 \cdot 3}{3 (1 + 3 x)}$

Multiply $3$ by $3$ .

$y = \frac{7 x + 9}{3 (1 + 3 x)}$

Step 3

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

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

Step 4

Verify if $f^{- 1} (x) = \frac{7 x + 9}{3 (1 + 3 x)}$ is the inverse of $f (x) = \frac{3 x - 9}{- 9 x + 7}$ .

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 $f^{- 1} (\frac{3 x - 9}{- 9 x + 7})$ by substituting in the value of $f$ into $f^{- 1}$ .

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

Simplify with factoring out.

Factor $3$ out of $3 x - 9$ .

Factor $3$ out of $3 x$ .

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

Factor $3$ out of $- 9$ .

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

Factor $3$ out of $3 x + 3 \cdot - 3$ .

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

Factor $3$ out of $3 x - 9$ .

Factor $3$ out of $3 x$ .

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

Factor $3$ out of $- 9$ .

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

Factor $3$ out of $3 x + 3 \cdot - 3$ .

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

Simplify the numerator.

Multiply $7 \frac{3 (x - 3)}{- 9 x + 7}$ .

Combine $7$ and $\frac{3 (x - 3)}{- 9 x + 7}$ .

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

Multiply $3$ by $7$ .

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

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

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

Combine the numerators over the common denominator.

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

Rewrite $\frac{21 (x - 3) + 9 (- 9 x + 7)}{- 9 x + 7}$ in a factored form.

Factor $3$ out of $21 (x - 3) + 9 (- 9 x + 7)$ .

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

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

Factor $3$ out of $9 (- 9 x + 7)$ .

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

Factor $3$ out of $3 (7 (x - 3)) + 3 (3 (- 9 x + 7))$ .

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

Apply the distributive property.

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

Multiply $7$ by $- 3$ .

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

Apply the distributive property.

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

Multiply $- 9$ by $3$ .

$f^{- 1} (\frac{3 x - 9}{- 9 x + 7}) = \frac{\frac{3 (7 x - 21 - 27 x + 3 \cdot 7)}{- 9 x + 7}}{3 (1 + 3 (\frac{3 (x - 3)}{- 9 x + 7}))}$

Multiply $3$ by $7$ .

$f^{- 1} (\frac{3 x - 9}{- 9 x + 7}) = \frac{\frac{3 (7 x - 21 - 27 x + 21)}{- 9 x + 7}}{3 (1 + 3 (\frac{3 (x - 3)}{- 9 x + 7}))}$

Subtract $27 x$ from $7 x$ .

$f^{- 1} (\frac{3 x - 9}{- 9 x + 7}) = \frac{\frac{3 (- 20 x - 21 + 21)}{- 9 x + 7}}{3 (1 + 3 (\frac{3 (x - 3)}{- 9 x + 7}))}$

Add $- 21$ and $21$ .

$f^{- 1} (\frac{3 x - 9}{- 9 x + 7}) = \frac{\frac{3 (- 20 x + 0)}{- 9 x + 7}}{3 (1 + 3 (\frac{3 (x - 3)}{- 9 x + 7}))}$

Add $- 20 x$ and $0$ .

$f^{- 1} (\frac{3 x - 9}{- 9 x + 7}) = \frac{\frac{3 \cdot (- 20 x)}{- 9 x + 7}}{3 (1 + 3 (\frac{3 (x - 3)}{- 9 x + 7}))}$

Multiply $3$ by $- 20$ .

$f^{- 1} (\frac{3 x - 9}{- 9 x + 7}) = \frac{\frac{- 60 x}{- 9 x + 7}}{3 (1 + 3 (\frac{3 (x - 3)}{- 9 x + 7}))}$

Move the negative in front of the fraction.

$f^{- 1} (\frac{3 x - 9}{- 9 x + 7}) = \frac{- \frac{60 x}{- 9 x + 7}}{3 (1 + 3 (\frac{3 (x - 3)}{- 9 x + 7}))}$

Simplify the denominator.

Multiply $3 \frac{3 (x - 3)}{- 9 x + 7}$ .

Combine $3$ and $\frac{3 (x - 3)}{- 9 x + 7}$ .

$f^{- 1} (\frac{3 x - 9}{- 9 x + 7}) = \frac{- \frac{60 x}{- 9 x + 7}}{3 (1 + \frac{3 (3 (x - 3))}{- 9 x + 7})}$

Multiply $3$ by $3$ .

$f^{- 1} (\frac{3 x - 9}{- 9 x + 7}) = \frac{- \frac{60 x}{- 9 x + 7}}{3 (1 + \frac{9 (x - 3)}{- 9 x + 7})}$

Write $1$ as a fraction with a common denominator.

$f^{- 1} (\frac{3 x - 9}{- 9 x + 7}) = \frac{- \frac{60 x}{- 9 x + 7}}{3 (\frac{- 9 x + 7}{- 9 x + 7} + \frac{9 (x - 3)}{- 9 x + 7})}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{3 x - 9}{- 9 x + 7}) = \frac{- \frac{60 x}{- 9 x + 7}}{3 (\frac{- 9 x + 7 + 9 (x - 3)}{- 9 x + 7})}$

Reorder terms.

$f^{- 1} (\frac{3 x - 9}{- 9 x + 7}) = \frac{- \frac{60 x}{- 9 x + 7}}{3 (\frac{- 9 x + 9 (x - 3) + 7}{- 9 x + 7})}$

Rewrite $\frac{- 9 x + 9 (x - 3) + 7}{- 9 x + 7}$ in a factored form.

Apply the distributive property.

$f^{- 1} (\frac{3 x - 9}{- 9 x + 7}) = \frac{- \frac{60 x}{- 9 x + 7}}{3 (\frac{- 9 x + 9 x + 9 \cdot - 3 + 7}{- 9 x + 7})}$

Multiply $9$ by $- 3$ .

$f^{- 1} (\frac{3 x - 9}{- 9 x + 7}) = \frac{- \frac{60 x}{- 9 x + 7}}{3 (\frac{- 9 x + 9 x - 27 + 7}{- 9 x + 7})}$

Add $- 9 x$ and $9 x$ .

$f^{- 1} (\frac{3 x - 9}{- 9 x + 7}) = \frac{- \frac{60 x}{- 9 x + 7}}{3 (\frac{0 - 27 + 7}{- 9 x + 7})}$

Subtract $27$ from $0$ .

$f^{- 1} (\frac{3 x - 9}{- 9 x + 7}) = \frac{- \frac{60 x}{- 9 x + 7}}{3 (\frac{- 27 + 7}{- 9 x + 7})}$

Add $- 27$ and $7$ .

$f^{- 1} (\frac{3 x - 9}{- 9 x + 7}) = \frac{- \frac{60 x}{- 9 x + 7}}{3 (\frac{- 20}{- 9 x + 7})}$

Move the negative in front of the fraction.

$f^{- 1} (\frac{3 x - 9}{- 9 x + 7}) = \frac{- \frac{60 x}{- 9 x + 7}}{3 \cdot (- 1 \frac{20}{- 9 x + 7})}$

Combine exponents.

Factor out negative.

$f^{- 1} (\frac{3 x - 9}{- 9 x + 7}) = \frac{- \frac{60 x}{- 9 x + 7}}{- (3 (\frac{20}{- 9 x + 7}))}$

Combine $3$ and $\frac{20}{- 9 x + 7}$ .

$f^{- 1} (\frac{3 x - 9}{- 9 x + 7}) = \frac{- \frac{60 x}{- 9 x + 7}}{- \frac{3 \cdot 20}{- 9 x + 7}}$

Multiply $3$ by $20$ .

$f^{- 1} (\frac{3 x - 9}{- 9 x + 7}) = \frac{- \frac{60 x}{- 9 x + 7}}{- \frac{60}{- 9 x + 7}}$

Dividing two negative values results in a positive value.

$f^{- 1} (\frac{3 x - 9}{- 9 x + 7}) = \frac{\frac{60 x}{- 9 x + 7}}{\frac{60}{- 9 x + 7}}$

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{3 x - 9}{- 9 x + 7}) = \frac{60 x}{- 9 x + 7} \cdot \frac{- 9 x + 7}{60}$

Cancel the common factor of $60$ .

Factor $60$ out of $60 x$ .

$f^{- 1} (\frac{3 x - 9}{- 9 x + 7}) = \frac{60 (x)}{- 9 x + 7} \cdot \frac{- 9 x + 7}{60}$

Cancel the common factor.

$f^{- 1} (\frac{3 x - 9}{- 9 x + 7}) = \frac{60 x}{- 9 x + 7} \cdot \frac{- 9 x + 7}{60}$

Rewrite the expression.

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

Multiply $\frac{x}{- 9 x + 7}$ by $- 9 x + 7$ .

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

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

Cancel the common factor.

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

Divide $x$ by $1$ .

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

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

Set up the composite result function.

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

Evaluate $f (\frac{7 x + 9}{3 (1 + 3 x)})$ by substituting in the value of $f^{- 1}$ into $f$ .

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

Simplify the numerator.

Factor $3$ out of $3 \frac{7 x + 9}{3 (1 + 3 x)} - 9$ .

Factor $3$ out of $3 \frac{7 x + 9}{3 (1 + 3 x)}$ .

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

Factor $3$ out of $- 9$ .

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

Factor $3$ out of $3 \frac{7 x + 9}{3 (1 + 3 x)} + 3 \cdot - 3$ .

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

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{3 (1 + 3 x)}{3 (1 + 3 x)}$ .

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

Combine $- 3$ and $\frac{3 (1 + 3 x)}{3 (1 + 3 x)}$ .

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

Combine the numerators over the common denominator.

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

Rewrite $\frac{7 x + 9 - 3 (3 (1 + 3 x))}{3 (1 + 3 x)}$ in a factored form.

Apply the distributive property.

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

Multiply $3$ by $1$ .

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

Multiply $3$ by $3$ .

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

Apply the distributive property.

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

Multiply $- 3$ by $3$ .

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

Multiply $9$ by $- 3$ .

$f (\frac{7 x + 9}{3 (1 + 3 x)}) = \frac{3 (\frac{7 x + 9 - 9 - 27 x}{3 (1 + 3 x)})}{- 9 \frac{7 x + 9}{3 (1 + 3 x)} + 7}$

Subtract $27 x$ from $7 x$ .

$f (\frac{7 x + 9}{3 (1 + 3 x)}) = \frac{3 (\frac{- 20 x + 9 - 9}{3 (1 + 3 x)})}{- 9 \frac{7 x + 9}{3 (1 + 3 x)} + 7}$

Subtract $9$ from $9$ .

$f (\frac{7 x + 9}{3 (1 + 3 x)}) = \frac{3 (\frac{- 20 x + 0}{3 (1 + 3 x)})}{- 9 \frac{7 x + 9}{3 (1 + 3 x)} + 7}$

Add $- 20 x$ and $0$ .

$f (\frac{7 x + 9}{3 (1 + 3 x)}) = \frac{3 (\frac{- 20 x}{3 (1 + 3 x)})}{- 9 \frac{7 x + 9}{3 (1 + 3 x)} + 7}$

Move the negative in front of the fraction.

$f (\frac{7 x + 9}{3 (1 + 3 x)}) = \frac{3 (- \frac{(20) x}{3 (1 + 3 x)})}{- 9 \frac{7 x + 9}{3 (1 + 3 x)} + 7}$

Remove unnecessary parentheses.

$f (\frac{7 x + 9}{3 (1 + 3 x)}) = \frac{3 \cdot (- 1 \frac{20 x}{3 (1 + 3 x)})}{- 9 \frac{7 x + 9}{3 (1 + 3 x)} + 7}$

Combine exponents.

Factor out negative.

$f (\frac{7 x + 9}{3 (1 + 3 x)}) = \frac{- (3 (\frac{20 x}{3 (1 + 3 x)}))}{- 9 \frac{7 x + 9}{3 (1 + 3 x)} + 7}$

Combine $3$ and $\frac{20 x}{3 (1 + 3 x)}$ .

$f (\frac{7 x + 9}{3 (1 + 3 x)}) = \frac{- \frac{3 (20 x)}{3 (1 + 3 x)}}{- 9 \frac{7 x + 9}{3 (1 + 3 x)} + 7}$

Multiply $20$ by $3$ .

$f (\frac{7 x + 9}{3 (1 + 3 x)}) = \frac{- \frac{60 x}{3 (1 + 3 x)}}{- 9 \frac{7 x + 9}{3 (1 + 3 x)} + 7}$

Reduce the expression $\frac{60 x}{3 (1 + 3 x)}$ by cancelling the common factors.

Factor $3$ out of $60 x$ .

$f (\frac{7 x + 9}{3 (1 + 3 x)}) = \frac{- \frac{3 (20 x)}{3 (1 + 3 x)}}{- 9 \frac{7 x + 9}{3 (1 + 3 x)} + 7}$

Cancel the common factor.

$f (\frac{7 x + 9}{3 (1 + 3 x)}) = \frac{- \frac{3 (20 x)}{3 (1 + 3 x)}}{- 9 \frac{7 x + 9}{3 (1 + 3 x)} + 7}$

Rewrite the expression.

$f (\frac{7 x + 9}{3 (1 + 3 x)}) = \frac{- \frac{20 x}{1 + 3 x}}{- 9 \frac{7 x + 9}{3 (1 + 3 x)} + 7}$

Simplify the denominator.

Cancel the common factor of $3$ .

Factor $3$ out of $- 9$ .

$f (\frac{7 x + 9}{3 (1 + 3 x)}) = \frac{- \frac{20 x}{1 + 3 x}}{3 (- 3) (\frac{7 x + 9}{3 (1 + 3 x)}) + 7}$

Cancel the common factor.

$f (\frac{7 x + 9}{3 (1 + 3 x)}) = \frac{- \frac{20 x}{1 + 3 x}}{3 \cdot (- 3 \frac{7 x + 9}{3 (1 + 3 x)}) + 7}$

Rewrite the expression.

$f (\frac{7 x + 9}{3 (1 + 3 x)}) = \frac{- \frac{20 x}{1 + 3 x}}{- 3 \frac{7 x + 9}{1 + 3 x} + 7}$

Combine $- 3$ and $\frac{7 x + 9}{1 + 3 x}$ .

$f (\frac{7 x + 9}{3 (1 + 3 x)}) = \frac{- \frac{20 x}{1 + 3 x}}{\frac{- 3 (7 x + 9)}{1 + 3 x} + 7}$

Move the negative in front of the fraction.

$f (\frac{7 x + 9}{3 (1 + 3 x)}) = \frac{- \frac{20 x}{1 + 3 x}}{- \frac{(3) (7 x + 9)}{1 + 3 x} + 7}$

To write $7$ as a fraction with a common denominator, multiply by $\frac{1 + 3 x}{1 + 3 x}$ .

$f (\frac{7 x + 9}{3 (1 + 3 x)}) = \frac{- \frac{20 x}{1 + 3 x}}{- \frac{3 (7 x + 9)}{1 + 3 x} + \frac{7 (1 + 3 x)}{1 + 3 x}}$

Combine the numerators over the common denominator.

$f (\frac{7 x + 9}{3 (1 + 3 x)}) = \frac{- \frac{20 x}{1 + 3 x}}{\frac{- 3 (7 x + 9) + 7 (1 + 3 x)}{1 + 3 x}}$

Rewrite $\frac{- 3 (7 x + 9) + 7 (1 + 3 x)}{1 + 3 x}$ in a factored form.

Apply the distributive property.

$f (\frac{7 x + 9}{3 (1 + 3 x)}) = \frac{- \frac{20 x}{1 + 3 x}}{\frac{- 3 (7 x) - 3 \cdot 9 + 7 (1 + 3 x)}{1 + 3 x}}$

Multiply $7$ by $- 3$ .

$f (\frac{7 x + 9}{3 (1 + 3 x)}) = \frac{- \frac{20 x}{1 + 3 x}}{\frac{- 21 x - 3 \cdot 9 + 7 (1 + 3 x)}{1 + 3 x}}$

Multiply $- 3$ by $9$ .

$f (\frac{7 x + 9}{3 (1 + 3 x)}) = \frac{- \frac{20 x}{1 + 3 x}}{\frac{- 21 x - 27 + 7 (1 + 3 x)}{1 + 3 x}}$

Apply the distributive property.

$f (\frac{7 x + 9}{3 (1 + 3 x)}) = \frac{- \frac{20 x}{1 + 3 x}}{\frac{- 21 x - 27 + 7 \cdot 1 + 7 (3 x)}{1 + 3 x}}$

Multiply $7$ by $1$ .

$f (\frac{7 x + 9}{3 (1 + 3 x)}) = \frac{- \frac{20 x}{1 + 3 x}}{\frac{- 21 x - 27 + 7 + 7 (3 x)}{1 + 3 x}}$

Multiply $3$ by $7$ .

$f (\frac{7 x + 9}{3 (1 + 3 x)}) = \frac{- \frac{20 x}{1 + 3 x}}{\frac{- 21 x - 27 + 7 + 21 x}{1 + 3 x}}$

Add $- 21 x$ and $21 x$ .

$f (\frac{7 x + 9}{3 (1 + 3 x)}) = \frac{- \frac{20 x}{1 + 3 x}}{\frac{0 - 27 + 7}{1 + 3 x}}$

Subtract $27$ from $0$ .

$f (\frac{7 x + 9}{3 (1 + 3 x)}) = \frac{- \frac{20 x}{1 + 3 x}}{\frac{- 27 + 7}{1 + 3 x}}$

Add $- 27$ and $7$ .

$f (\frac{7 x + 9}{3 (1 + 3 x)}) = \frac{- \frac{20 x}{1 + 3 x}}{\frac{- 20}{1 + 3 x}}$

Move the negative in front of the fraction.

$f (\frac{7 x + 9}{3 (1 + 3 x)}) = \frac{- \frac{20 x}{1 + 3 x}}{- \frac{20}{1 + 3 x}}$

Dividing two negative values results in a positive value.

$f (\frac{7 x + 9}{3 (1 + 3 x)}) = \frac{\frac{20 x}{1 + 3 x}}{\frac{20}{1 + 3 x}}$

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{7 x + 9}{3 (1 + 3 x)}) = \frac{20 x}{1 + 3 x} \cdot \frac{1 + 3 x}{20}$

Cancel the common factor of $20$ .

Factor $20$ out of $20 x$ .

$f (\frac{7 x + 9}{3 (1 + 3 x)}) = \frac{20 (x)}{1 + 3 x} \cdot \frac{1 + 3 x}{20}$

Cancel the common factor.

$f (\frac{7 x + 9}{3 (1 + 3 x)}) = \frac{20 x}{1 + 3 x} \cdot \frac{1 + 3 x}{20}$

Rewrite the expression.

$f (\frac{7 x + 9}{3 (1 + 3 x)}) = \frac{x}{1 + 3 x} \cdot (1 + 3 x)$

Cancel the common factor of $1 + 3 x$ .

Cancel the common factor.

$f (\frac{7 x + 9}{3 (1 + 3 x)}) = \frac{x}{1 + 3 x} \cdot (1 + 3 x)$

Rewrite the expression.

$f (\frac{7 x + 9}{3 (1 + 3 x)}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{7 x + 9}{3 (1 + 3 x)}$ is the inverse of $f (x) = \frac{3 x - 9}{- 9 x + 7}$ .

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

Find the Inverse (3x-9)/(-9x+7)

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