Algebra Examples

$\frac{- 2 x - 9}{- 6 x - 7}$

Step 1

Interchange the variables.

$x = \frac{- 2 y - 9}{- 6 y - 7}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{- 2 y - 9}{- 6 y - 7} = x$ .

$\frac{- 2 y - 9}{- 6 y - 7} = x$

Step 2.2

Find the LCD of the terms in the equation.

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Step 2.2.1

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

$- 6 y - 7, 1$

Step 2.2.2

Remove parentheses.

$- 6 y - 7, 1$

Step 2.2.3

The LCM of one and any expression is the expression.

$- 6 y - 7$

Step 2.3

Multiply each term in $\frac{- 2 y - 9}{- 6 y - 7} = x$ by $- 6 y - 7$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $\frac{- 2 y - 9}{- 6 y - 7} = x$ by $- 6 y - 7$ .

$\frac{- 2 y - 9}{- 6 y - 7} (- 6 y - 7) = x (- 6 y - 7)$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Multiply $\frac{- 2 y - 9}{- 6 y - 7}$ by $- 6 y - 7$ .

$\frac{(- 2 y - 9) (- 6 y - 7)}{- 6 y - 7} = x (- 6 y - 7)$

Step 2.3.2.2

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

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Step 2.3.2.2.1

Cancel the common factor.

$\frac{(- 2 y - 9) (- 6 y - 7)}{- 6 y - 7} = x (- 6 y - 7)$

Step 2.3.2.2.2

Divide $- 2 y - 9$ by $1$ .

$- 2 y - 9 = x (- 6 y - 7)$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Apply the distributive property.

$- 2 y - 9 = x (- 6 y) + x \cdot - 7$

Step 2.3.3.2

Reorder.

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Step 2.3.3.2.1

Rewrite using the commutative property of multiplication.

$- 2 y - 9 = - 6 x y + x \cdot - 7$

Step 2.3.3.2.2

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

$- 2 y - 9 = - 6 x y - 7 x$

Step 2.4

Solve the equation.

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Step 2.4.1

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

$- 2 y - 9 + 6 x y = - 7 x$

Step 2.4.2

Add $9$ to both sides of the equation.

$- 2 y + 6 x y = - 7 x + 9$

Step 2.4.3

Factor $2 y$ out of $- 2 y + 6 x y$ .

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Step 2.4.3.1

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

$2 y (- 1) + 6 x y = - 7 x + 9$

Step 2.4.3.2

Factor $2 y$ out of $6 x y$ .

$2 y (- 1) + 2 y (3 x) = - 7 x + 9$

Step 2.4.3.3

Factor $2 y$ out of $2 y (- 1) + 2 y (3 x)$ .

$2 y (- 1 + 3 x) = - 7 x + 9$

Step 2.4.4

Divide each term in $2 y (- 1 + 3 x) = - 7 x + 9$ by $2 (- 1 + 3 x)$ and simplify.

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Step 2.4.4.1

Divide each term in $2 y (- 1 + 3 x) = - 7 x + 9$ by $2 (- 1 + 3 x)$ .

$\frac{2 y (- 1 + 3 x)}{2 (- 1 + 3 x)} = \frac{- 7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}$

Step 2.4.4.2

Simplify the left side.

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Step 2.4.4.2.1

Cancel the common factor of $2$ .

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Step 2.4.4.2.1.1

Cancel the common factor.

$\frac{2 y (- 1 + 3 x)}{2 (- 1 + 3 x)} = \frac{- 7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}$

Step 2.4.4.2.1.2

Rewrite the expression.

$\frac{y (- 1 + 3 x)}{- 1 + 3 x} = \frac{- 7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}$

Step 2.4.4.2.2

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

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Step 2.4.4.2.2.1

Cancel the common factor.

$\frac{y (- 1 + 3 x)}{- 1 + 3 x} = \frac{- 7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}$

Step 2.4.4.2.2.2

Divide $y$ by $1$ .

$y = \frac{- 7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}$

Step 2.4.4.3

Simplify the right side.

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Step 2.4.4.3.1

Move the negative in front of the fraction.

$y = - \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}$

Step 3

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

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

Step 4

Verify if $f^{- 1} (x) = - \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}$ is the inverse of $f (x) = \frac{- 2 x - 9}{- 6 x - 7}$ .

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Step 4.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

Evaluate $f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7})$ by substituting in the value of $f$ into $f^{- 1}$ .

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

Step 4.2.3

Combine the numerators over the common denominator.

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

Step 4.2.4

Simplify each term.

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Step 4.2.4.1

Combine $- 7$ and $\frac{- 2 x - 9}{- 6 x - 7}$ .

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

Step 4.2.4.2

Move the negative in front of the fraction.

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

Step 4.2.5

To write $9$ as a fraction with a common denominator, multiply by $\frac{- 6 x - 7}{- 6 x - 7}$ .

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

Step 4.2.6

Combine the numerators over the common denominator.

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

Step 4.2.7

Simplify the numerator.

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Step 4.2.7.1

Apply the distributive property.

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

Step 4.2.7.2

Multiply $- 2$ by $- 7$ .

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

Step 4.2.7.3

Multiply $- 7$ by $- 9$ .

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{\frac{14 x + 63 + 9 (- 6 x - 7)}{- 6 x - 7}}{2 (- 1 + 3 (\frac{- 2 x - 9}{- 6 x - 7}))}$

Step 4.2.7.4

Apply the distributive property.

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{\frac{14 x + 63 + 9 (- 6 x) + 9 \cdot - 7}{- 6 x - 7}}{2 (- 1 + 3 (\frac{- 2 x - 9}{- 6 x - 7}))}$

Step 4.2.7.5

Multiply $- 6$ by $9$ .

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{\frac{14 x + 63 - 54 x + 9 \cdot - 7}{- 6 x - 7}}{2 (- 1 + 3 (\frac{- 2 x - 9}{- 6 x - 7}))}$

Step 4.2.7.6

Multiply $9$ by $- 7$ .

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{\frac{14 x + 63 - 54 x - 63}{- 6 x - 7}}{2 (- 1 + 3 (\frac{- 2 x - 9}{- 6 x - 7}))}$

Step 4.2.7.7

Subtract $54 x$ from $14 x$ .

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{\frac{- 40 x + 63 - 63}{- 6 x - 7}}{2 (- 1 + 3 (\frac{- 2 x - 9}{- 6 x - 7}))}$

Step 4.2.7.8

Subtract $63$ from $63$ .

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{\frac{- 40 x + 0}{- 6 x - 7}}{2 (- 1 + 3 (\frac{- 2 x - 9}{- 6 x - 7}))}$

Step 4.2.7.9

Add $- 40 x$ and $0$ .

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{\frac{- 40 x}{- 6 x - 7}}{2 (- 1 + 3 (\frac{- 2 x - 9}{- 6 x - 7}))}$

Step 4.2.8

Simplify with factoring out.

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Step 4.2.8.1

Move the negative in front of the fraction.

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{- \frac{40 x}{- 6 x - 7}}{2 (- 1 + 3 (\frac{- 2 x - 9}{- 6 x - 7}))}$

Step 4.2.8.2

Factor $- 1$ out of $- 6 x$ .

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{- \frac{40 x}{- (6 x) - 7}}{2 (- 1 + 3 (\frac{- 2 x - 9}{- 6 x - 7}))}$

Step 4.2.8.3

Rewrite $- 7$ as $- 1 (7)$ .

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{- \frac{40 x}{- (6 x) - 1 \cdot 7}}{2 (- 1 + 3 (\frac{- 2 x - 9}{- 6 x - 7}))}$

Step 4.2.8.4

Factor $- 1$ out of $- (6 x) - 1 (7)$ .

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{- \frac{40 x}{- (6 x + 7)}}{2 (- 1 + 3 (\frac{- 2 x - 9}{- 6 x - 7}))}$

Step 4.2.8.5

Simplify the expression.

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Step 4.2.8.5.1

Rewrite $- (6 x + 7)$ as $- 1 (6 x + 7)$ .

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{- \frac{40 x}{- 1 (6 x + 7)}}{2 (- 1 + 3 (\frac{- 2 x - 9}{- 6 x - 7}))}$

Step 4.2.8.5.2

Move the negative in front of the fraction.

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{\frac{40 x}{6 x + 7}}{2 (- 1 + 3 (\frac{- 2 x - 9}{- 6 x - 7}))}$

Step 4.2.8.5.3

Multiply $- 1$ by $- 1$ .

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{1 (\frac{40 x}{6 x + 7})}{2 (- 1 + 3 (\frac{- 2 x - 9}{- 6 x - 7}))}$

Step 4.2.8.5.4

Multiply $\frac{40 x}{6 x + 7}$ by $1$ .

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{\frac{40 x}{6 x + 7}}{2 (- 1 + 3 (\frac{- 2 x - 9}{- 6 x - 7}))}$

Step 4.2.9

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{40 x}{6 x + 7} \cdot \frac{1}{2 (- 1 + 3 (\frac{- 2 x - 9}{- 6 x - 7}))}$

Step 4.2.10

Simplify terms.

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Step 4.2.10.1

Cancel the common factor of $2$ .

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Step 4.2.10.1.1

Factor $2$ out of $40 x$ .

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{2 (20 x)}{6 x + 7} \cdot \frac{1}{2 (- 1 + 3 (\frac{- 2 x - 9}{- 6 x - 7}))}$

Step 4.2.10.1.2

Cancel the common factor.

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{2 (20 x)}{6 x + 7} \cdot \frac{1}{2 (- 1 + 3 (\frac{- 2 x - 9}{- 6 x - 7}))}$

Step 4.2.10.1.3

Rewrite the expression.

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{20 x}{6 x + 7} \cdot \frac{1}{- 1 + 3 (\frac{- 2 x - 9}{- 6 x - 7})}$

Step 4.2.10.2

Combine $3$ and $\frac{- 2 x - 9}{- 6 x - 7}$ .

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{20 x}{6 x + 7} \cdot \frac{1}{- 1 + \frac{3 (- 2 x - 9)}{- 6 x - 7}}$

Step 4.2.10.3

Multiply $\frac{20 x}{6 x + 7}$ by $\frac{1}{- 1 + \frac{3 (- 2 x - 9)}{- 6 x - 7}}$ .

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{20 x}{(6 x + 7) (- 1 + \frac{3 (- 2 x - 9)}{- 6 x - 7})}$

Step 4.2.11

Simplify the denominator.

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Step 4.2.11.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{- 6 x - 7}{- 6 x - 7}$ .

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{20 x}{(6 x + 7) (- 1 \cdot \frac{- 6 x - 7}{- 6 x - 7} + \frac{3 (- 2 x - 9)}{- 6 x - 7})}$

Step 4.2.11.2

Combine $- 1$ and $\frac{- 6 x - 7}{- 6 x - 7}$ .

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{20 x}{(6 x + 7) (\frac{- (- 6 x - 7)}{- 6 x - 7} + \frac{3 (- 2 x - 9)}{- 6 x - 7})}$

Step 4.2.11.3

Combine the numerators over the common denominator.

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{20 x}{(6 x + 7) (\frac{- (- 6 x - 7) + 3 (- 2 x - 9)}{- 6 x - 7})}$

Step 4.2.11.4

Reorder terms.

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{20 x}{(6 x + 7) (\frac{3 (- 2 x - 9) - (- 6 x - 7)}{- 6 x - 7})}$

Step 4.2.11.5

Rewrite $\frac{3 (- 2 x - 9) - (- 6 x - 7)}{- 6 x - 7}$ in a factored form.

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Step 4.2.11.5.1

Apply the distributive property.

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{20 x}{(6 x + 7) (\frac{3 (- 2 x) + 3 \cdot - 9 - (- 6 x - 7)}{- 6 x - 7})}$

Step 4.2.11.5.2

Multiply $- 2$ by $3$ .

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{20 x}{(6 x + 7) (\frac{- 6 x + 3 \cdot - 9 - (- 6 x - 7)}{- 6 x - 7})}$

Step 4.2.11.5.3

Multiply $3$ by $- 9$ .

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{20 x}{(6 x + 7) (\frac{- 6 x - 27 - (- 6 x - 7)}{- 6 x - 7})}$

Step 4.2.11.5.4

Apply the distributive property.

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{20 x}{(6 x + 7) (\frac{- 6 x - 27 - (- 6 x) + 7}{- 6 x - 7})}$

Step 4.2.11.5.5

Multiply $- 6$ by $- 1$ .

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{20 x}{(6 x + 7) (\frac{- 6 x - 27 + 6 x + 7}{- 6 x - 7})}$

Step 4.2.11.5.6

Multiply $- 1$ by $- 7$ .

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{20 x}{(6 x + 7) (\frac{- 6 x - 27 + 6 x + 7}{- 6 x - 7})}$

Step 4.2.11.5.7

Add $- 6 x$ and $6 x$ .

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{20 x}{(6 x + 7) (\frac{0 - 27 + 7}{- 6 x - 7})}$

Step 4.2.11.5.8

Subtract $27$ from $0$ .

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{20 x}{(6 x + 7) (\frac{- 27 + 7}{- 6 x - 7})}$

Step 4.2.11.5.9

Add $- 27$ and $7$ .

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{20 x}{(6 x + 7) (\frac{- 20}{- 6 x - 7})}$

Step 4.2.11.6

Move the negative in front of the fraction.

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{20 x}{(6 x + 7) \cdot (- 1 \frac{20}{- 6 x - 7})}$

Step 4.2.11.7

Factor out negative.

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{20 x}{- (6 x + 7) \frac{20}{- 6 x - 7}}$

Step 4.2.12

Factor $\frac{20}{- 6 x - 7}$ out of $\frac{20 x}{- (6 x + 7) \frac{20}{- 6 x - 7}}$ .

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{- 6 x - 7}{20} \cdot \frac{20 x}{- (6 x + 7)}$

Step 4.2.13

Cancel the common factor of $20$ .

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Step 4.2.13.1

Factor $20$ out of $20 x$ .

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{- 6 x - 7}{20} \cdot \frac{20 (x)}{- (6 x + 7)}$

Step 4.2.13.2

Cancel the common factor.

$f^{- 1} (\frac{- 2 x - 9}{- 6 x - 7}) = \frac{- 6 x - 7}{20} \cdot \frac{20 x}{- (6 x + 7)}$

Step 4.2.13.3

Rewrite the expression.

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

Step 4.2.14

Move the negative in front of the fraction.

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

Step 4.2.15

Apply the distributive property.

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

Step 4.2.16

Multiply $- 6 x (- \frac{x}{6 x + 7})$ .

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Step 4.2.16.1

Multiply $- 1$ by $- 6$ .

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

Step 4.2.16.2

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

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

Step 4.2.16.3

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

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

Step 4.2.16.4

Raise $x$ to the power of $1$ .

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

Step 4.2.16.5

Raise $x$ to the power of $1$ .

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

Step 4.2.16.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.2.16.7

Add $1$ and $1$ .

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

Step 4.2.17

Multiply $- 7 (- \frac{x}{6 x + 7})$ .

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Step 4.2.17.1

Multiply $- 1$ by $- 7$ .

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

Step 4.2.17.2

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

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

Step 4.2.18

Combine the numerators over the common denominator.

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

Step 4.2.19

Factor $x$ out of $6 x^{2} + 7 x$ .

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Step 4.2.19.1

Factor $x$ out of $6 x^{2}$ .

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

Step 4.2.19.2

Factor $x$ out of $7 x$ .

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

Step 4.2.19.3

Factor $x$ out of $x (6 x) + x \cdot 7$ .

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

Step 4.2.20

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

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Step 4.2.20.1

Cancel the common factor.

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

Step 4.2.20.2

Divide $x$ by $1$ .

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

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

Evaluate $f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)})$ by substituting in the value of $f^{- 1}$ into $f$ .

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

Step 4.3.3

Simplify the numerator.

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Step 4.3.3.1

Factor $- 1$ out of $- 2 (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) - 9$ .

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Step 4.3.3.1.1

Reorder the expression.

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Step 4.3.3.1.1.1

Reorder $- 1$ and $3 x$ .

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

Step 4.3.3.1.1.2

Reorder $- 1$ and $3 x$ .

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

Step 4.3.3.1.2

Factor $- 1$ out of $- 2 (- \frac{7 x}{2 (3 x - 1)} + \frac{9}{2 (3 x - 1)})$ .

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

Step 4.3.3.1.3

Rewrite $- 9$ as $- 1 (9)$ .

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

Step 4.3.3.1.4

Factor $- 1$ out of $- (2 (- \frac{7 x}{2 (3 x - 1)} + \frac{9}{2 (3 x - 1)})) - 1 (9)$ .

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

Step 4.3.3.2

Apply the distributive property.

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

Step 4.3.3.3

Cancel the common factor of $2$ .

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Step 4.3.3.3.1

Move the leading negative in $- \frac{7 x}{2 (3 x - 1)}$ into the numerator.

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

Step 4.3.3.3.2

Cancel the common factor.

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

Step 4.3.3.3.3

Rewrite the expression.

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

Step 4.3.3.4

Cancel the common factor of $2$ .

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Step 4.3.3.4.1

Cancel the common factor.

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

Step 4.3.3.4.2

Rewrite the expression.

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

Step 4.3.3.5

Combine the numerators over the common denominator.

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

Step 4.3.3.6

To write $9$ as a fraction with a common denominator, multiply by $\frac{3 x - 1}{3 x - 1}$ .

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

Step 4.3.3.7

Combine the numerators over the common denominator.

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

Step 4.3.3.8

Reorder terms.

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

Step 4.3.3.9

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

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Step 4.3.3.9.1

Apply the distributive property.

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

Step 4.3.3.9.2

Multiply $3$ by $9$ .

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{- 7 x + 27 x + 9 \cdot - 1 + 9}{3 x - 1}}{- 6 (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) - 7}$

Step 4.3.3.9.3

Multiply $9$ by $- 1$ .

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{- 7 x + 27 x - 9 + 9}{3 x - 1}}{- 6 (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) - 7}$

Step 4.3.3.9.4

Add $- 7 x$ and $27 x$ .

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{20 x - 9 + 9}{3 x - 1}}{- 6 (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) - 7}$

Step 4.3.3.9.5

Add $- 9$ and $9$ .

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{20 x + 0}{3 x - 1}}{- 6 (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) - 7}$

Step 4.3.3.9.6

Add $20 x$ and $0$ .

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{20 x}{3 x - 1}}{- 6 (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) - 7}$

Step 4.3.4

Simplify the denominator.

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Step 4.3.4.1

Factor $- 1$ out of $- 6 (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) - 7$ .

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Step 4.3.4.1.1

Reorder the expression.

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Step 4.3.4.1.1.1

Reorder $- 1$ and $3 x$ .

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{20 x}{3 x - 1}}{- 6 (- \frac{7 x}{2 (3 x - 1)} + \frac{9}{2 (- 1 + 3 x)}) - 7}$

Step 4.3.4.1.1.2

Reorder $- 1$ and $3 x$ .

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{20 x}{3 x - 1}}{- 6 (- \frac{7 x}{2 (3 x - 1)} + \frac{9}{2 (3 x - 1)}) - 7}$

Step 4.3.4.1.2

Factor $- 1$ out of $- 6 (- \frac{7 x}{2 (3 x - 1)} + \frac{9}{2 (3 x - 1)})$ .

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{20 x}{3 x - 1}}{- (6 (- \frac{7 x}{2 (3 x - 1)} + \frac{9}{2 (3 x - 1)})) - 7}$

Step 4.3.4.1.3

Rewrite $- 7$ as $- 1 (7)$ .

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{20 x}{3 x - 1}}{- (6 (- \frac{7 x}{2 (3 x - 1)} + \frac{9}{2 (3 x - 1)})) - 1 \cdot 7}$

Step 4.3.4.1.4

Factor $- 1$ out of $- (6 (- \frac{7 x}{2 (3 x - 1)} + \frac{9}{2 (3 x - 1)})) - 1 (7)$ .

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{20 x}{3 x - 1}}{- (6 (- \frac{7 x}{2 (3 x - 1)} + \frac{9}{2 (3 x - 1)}) + 7)}$

Step 4.3.4.2

Apply the distributive property.

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{20 x}{3 x - 1}}{- (6 (- \frac{7 x}{2 (3 x - 1)}) + 6 (\frac{9}{2 (3 x - 1)}) + 7)}$

Step 4.3.4.3

Cancel the common factor of $2$ .

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Step 4.3.4.3.1

Move the leading negative in $- \frac{7 x}{2 (3 x - 1)}$ into the numerator.

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{20 x}{3 x - 1}}{- (6 (\frac{- 7 x}{2 (3 x - 1)}) + 6 (\frac{9}{2 (3 x - 1)}) + 7)}$

Step 4.3.4.3.2

Factor $2$ out of $6$ .

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{20 x}{3 x - 1}}{- (2 (3) (\frac{- 7 x}{2 (3 x - 1)}) + 6 (\frac{9}{2 (3 x - 1)}) + 7)}$

Step 4.3.4.3.3

Cancel the common factor.

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{20 x}{3 x - 1}}{- (2 \cdot (3 (\frac{- 7 x}{2 (3 x - 1)})) + 6 (\frac{9}{2 (3 x - 1)}) + 7)}$

Step 4.3.4.3.4

Rewrite the expression.

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{20 x}{3 x - 1}}{- (3 (\frac{- 7 x}{3 x - 1}) + 6 (\frac{9}{2 (3 x - 1)}) + 7)}$

Step 4.3.4.4

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

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{20 x}{3 x - 1}}{- (\frac{3 (- 7 x)}{3 x - 1} + 6 (\frac{9}{2 (3 x - 1)}) + 7)}$

Step 4.3.4.5

Multiply $- 7$ by $3$ .

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{20 x}{3 x - 1}}{- (\frac{- 21 x}{3 x - 1} + 6 (\frac{9}{2 (3 x - 1)}) + 7)}$

Step 4.3.4.6

Cancel the common factor of $2$ .

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Step 4.3.4.6.1

Factor $2$ out of $6$ .

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{20 x}{3 x - 1}}{- (\frac{- 21 x}{3 x - 1} + 2 (3) (\frac{9}{2 (3 x - 1)}) + 7)}$

Step 4.3.4.6.2

Cancel the common factor.

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{20 x}{3 x - 1}}{- (\frac{- 21 x}{3 x - 1} + 2 \cdot (3 (\frac{9}{2 (3 x - 1)})) + 7)}$

Step 4.3.4.6.3

Rewrite the expression.

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{20 x}{3 x - 1}}{- (\frac{- 21 x}{3 x - 1} + 3 (\frac{9}{3 x - 1}) + 7)}$

Step 4.3.4.7

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

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{20 x}{3 x - 1}}{- (\frac{- 21 x}{3 x - 1} + \frac{3 \cdot 9}{3 x - 1} + 7)}$

Step 4.3.4.8

Multiply $3$ by $9$ .

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{20 x}{3 x - 1}}{- (\frac{- 21 x}{3 x - 1} + \frac{27}{3 x - 1} + 7)}$

Step 4.3.4.9

Combine the numerators over the common denominator.

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{20 x}{3 x - 1}}{- (\frac{- 21 x + 27}{3 x - 1} + 7)}$

Step 4.3.4.10

Factor $3$ out of $- 21 x + 27$ .

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Step 4.3.4.10.1

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

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{20 x}{3 x - 1}}{- (\frac{3 (- 7 x) + 27}{3 x - 1} + 7)}$

Step 4.3.4.10.2

Factor $3$ out of $27$ .

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{20 x}{3 x - 1}}{- (\frac{3 (- 7 x) + 3 (9)}{3 x - 1} + 7)}$

Step 4.3.4.10.3

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

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{20 x}{3 x - 1}}{- (\frac{3 (- 7 x + 9)}{3 x - 1} + 7)}$

Step 4.3.4.11

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

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{20 x}{3 x - 1}}{- (\frac{3 (- 7 x + 9)}{3 x - 1} + \frac{7 (3 x - 1)}{3 x - 1})}$

Step 4.3.4.12

Combine the numerators over the common denominator.

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{20 x}{3 x - 1}}{- \frac{3 (- 7 x + 9) + 7 (3 x - 1)}{3 x - 1}}$

Step 4.3.4.13

Reorder terms.

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{20 x}{3 x - 1}}{- \frac{7 (3 x - 1) + 3 (- 7 x + 9)}{3 x - 1}}$

Step 4.3.4.14

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

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Step 4.3.4.14.1

Apply the distributive property.

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{20 x}{3 x - 1}}{- \frac{7 (3 x) + 7 \cdot - 1 + 3 (- 7 x + 9)}{3 x - 1}}$

Step 4.3.4.14.2

Multiply $3$ by $7$ .

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{20 x}{3 x - 1}}{- \frac{21 x + 7 \cdot - 1 + 3 (- 7 x + 9)}{3 x - 1}}$

Step 4.3.4.14.3

Multiply $7$ by $- 1$ .

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{20 x}{3 x - 1}}{- \frac{21 x - 7 + 3 (- 7 x + 9)}{3 x - 1}}$

Step 4.3.4.14.4

Apply the distributive property.

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{20 x}{3 x - 1}}{- \frac{21 x - 7 + 3 (- 7 x) + 3 \cdot 9}{3 x - 1}}$

Step 4.3.4.14.5

Multiply $- 7$ by $3$ .

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{20 x}{3 x - 1}}{- \frac{21 x - 7 - 21 x + 3 \cdot 9}{3 x - 1}}$

Step 4.3.4.14.6

Multiply $3$ by $9$ .

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{20 x}{3 x - 1}}{- \frac{21 x - 7 - 21 x + 27}{3 x - 1}}$

Step 4.3.4.14.7

Subtract $21 x$ from $21 x$ .

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{20 x}{3 x - 1}}{- \frac{0 - 7 + 27}{3 x - 1}}$

Step 4.3.4.14.8

Subtract $7$ from $0$ .

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{20 x}{3 x - 1}}{- \frac{- 7 + 27}{3 x - 1}}$

Step 4.3.4.14.9

Add $- 7$ and $27$ .

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{- \frac{20 x}{3 x - 1}}{- \frac{20}{3 x - 1}}$

Step 4.3.5

Dividing two negative values results in a positive value.

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{\frac{20 x}{3 x - 1}}{\frac{20}{3 x - 1}}$

Step 4.3.6

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{20 x}{3 x - 1} \cdot \frac{3 x - 1}{20}$

Step 4.3.7

Cancel the common factor of $20$ .

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Step 4.3.7.1

Factor $20$ out of $20 x$ .

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{20 (x)}{3 x - 1} \cdot \frac{3 x - 1}{20}$

Step 4.3.7.2

Cancel the common factor.

$f (- \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}) = \frac{20 x}{3 x - 1} \cdot \frac{3 x - 1}{20}$

Step 4.3.7.3

Rewrite the expression.

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

Step 4.3.8

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

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Step 4.3.8.1

Cancel the common factor.

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

Step 4.3.8.2

Rewrite the expression.

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

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{7 x}{2 (- 1 + 3 x)} + \frac{9}{2 (- 1 + 3 x)}$ is the inverse of $f (x) = \frac{- 2 x - 9}{- 6 x - 7}$ .

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