Precalculus Examples

$f (x) = \frac{- 8 x + 6}{- 9 x - 6}$

Step 1

Write $f (x) = \frac{- 8 x + 6}{- 9 x - 6}$ as an equation.

$y = \frac{- 8 x + 6}{- 9 x - 6}$

Step 2

Interchange the variables.

$x = \frac{- 8 y + 6}{- 9 y - 6}$

Step 3

Solve for $y$ .

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Step 3.1

Multiply the equation by $- 9 y - 6$ .

$(- 9 y - 6) (x) = (\frac{- 8 y + 6}{- 9 y - 6}) (- 9 y - 6)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Apply the distributive property.

$- 9 y x - 6 x = (\frac{- 8 y + 6}{- 9 y - 6}) (- 9 y - 6)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify $(\frac{- 8 y + 6}{- 9 y - 6}) (- 9 y - 6)$ .

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Step 3.3.1.1

Factor $2$ out of $- 8 y + 6$ .

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Step 3.3.1.1.1

Factor $2$ out of $- 8 y$ .

$- 9 y x - 6 x = \frac{2 (- 4 y) + 6}{- 9 y - 6} (- 9 y - 6)$

Step 3.3.1.1.2

Factor $2$ out of $6$ .

$- 9 y x - 6 x = \frac{2 (- 4 y) + 2 (3)}{- 9 y - 6} (- 9 y - 6)$

Step 3.3.1.1.3

Factor $2$ out of $2 (- 4 y) + 2 (3)$ .

$- 9 y x - 6 x = \frac{2 (- 4 y + 3)}{- 9 y - 6} (- 9 y - 6)$

Step 3.3.1.2

Factor $3$ out of $- 9 y - 6$ .

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Step 3.3.1.2.1

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

$- 9 y x - 6 x = \frac{2 (- 4 y + 3)}{3 (- 3 y) - 6} (- 9 y - 6)$

Step 3.3.1.2.2

Factor $3$ out of $- 6$ .

$- 9 y x - 6 x = \frac{2 (- 4 y + 3)}{3 (- 3 y) + 3 (- 2)} (- 9 y - 6)$

Step 3.3.1.2.3

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

$- 9 y x - 6 x = \frac{2 (- 4 y + 3)}{3 (- 3 y - 2)} (- 9 y - 6)$

Step 3.3.1.3

Multiply $\frac{2 (- 4 y + 3)}{3 (- 3 y - 2)}$ by $- 9 y - 6$ .

$- 9 y x - 6 x = \frac{2 (- 4 y + 3) (- 9 y - 6)}{3 (- 3 y - 2)}$

Step 3.3.1.4

Cancel the common factor of $- 9 y - 6$ and $3$ .

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Step 3.3.1.4.1

Factor $3$ out of $2 (- 4 y + 3) (- 9 y - 6)$ .

$- 9 y x - 6 x = \frac{3 (2 (- 4 y + 3) (- 3 y - 2))}{3 (- 3 y - 2)}$

Step 3.3.1.4.2

Cancel the common factors.

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Step 3.3.1.4.2.1

Cancel the common factor.

$- 9 y x - 6 x = \frac{3 (2 (- 4 y + 3) (- 3 y - 2))}{3 (- 3 y - 2)}$

Step 3.3.1.4.2.2

Rewrite the expression.

$- 9 y x - 6 x = \frac{2 (- 4 y + 3) (- 3 y - 2)}{- 3 y - 2}$

Step 3.3.1.5

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

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Step 3.3.1.5.1

Cancel the common factor.

$- 9 y x - 6 x = \frac{2 (- 4 y + 3) (- 3 y - 2)}{- 3 y - 2}$

Step 3.3.1.5.2

Divide $2 (- 4 y + 3)$ by $1$ .

$- 9 y x - 6 x = 2 (- 4 y + 3)$

Step 3.3.1.6

Apply the distributive property.

$- 9 y x - 6 x = 2 (- 4 y) + 2 \cdot 3$

Step 3.3.1.7

Multiply.

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Step 3.3.1.7.1

Multiply $- 4$ by $2$ .

$- 9 y x - 6 x = - 8 y + 2 \cdot 3$

Step 3.3.1.7.2

Multiply $2$ by $3$ .

$- 9 y x - 6 x = - 8 y + 6$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Add $8 y$ to both sides of the equation.

$- 9 y x - 6 x + 8 y = 6$

Step 3.4.2

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

$- 9 y x + 8 y = 6 + 6 x$

Step 3.4.3

Factor $y$ out of $- 9 y x + 8 y$ .

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Step 3.4.3.1

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

$y (- 9 x) + 8 y = 6 + 6 x$

Step 3.4.3.2

Factor $y$ out of $8 y$ .

$y (- 9 x) + y \cdot 8 = 6 + 6 x$

Step 3.4.3.3

Factor $y$ out of $y (- 9 x) + y \cdot 8$ .

$y (- 9 x + 8) = 6 + 6 x$

Step 3.4.4

Divide each term in $y (- 9 x + 8) = 6 + 6 x$ by $- 9 x + 8$ and simplify.

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Step 3.4.4.1

Divide each term in $y (- 9 x + 8) = 6 + 6 x$ by $- 9 x + 8$ .

$\frac{y (- 9 x + 8)}{- 9 x + 8} = \frac{6}{- 9 x + 8} + \frac{6 x}{- 9 x + 8}$

Step 3.4.4.2

Simplify the left side.

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Step 3.4.4.2.1

Cancel the common factor of $- 9 x + 8$ .

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Step 3.4.4.2.1.1

Cancel the common factor.

$\frac{y (- 9 x + 8)}{- 9 x + 8} = \frac{6}{- 9 x + 8} + \frac{6 x}{- 9 x + 8}$

Step 3.4.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{6}{- 9 x + 8} + \frac{6 x}{- 9 x + 8}$

Step 3.4.4.3

Simplify the right side.

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Step 3.4.4.3.1

Combine the numerators over the common denominator.

$y = \frac{6 + 6 x}{- 9 x + 8}$

Step 3.4.4.3.2

Factor $6$ out of $6 + 6 x$ .

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Step 3.4.4.3.2.1

Factor $6$ out of $6$ .

$y = \frac{6 \cdot 1 + 6 x}{- 9 x + 8}$

Step 3.4.4.3.2.2

Factor $6$ out of $6 \cdot 1 + 6 x$ .

$y = \frac{6 (1 + x)}{- 9 x + 8}$

Step 3.4.4.3.3

Factor $- 1$ out of $- 9 x$ .

$y = \frac{6 (1 + x)}{- (9 x) + 8}$

Step 3.4.4.3.4

Rewrite $8$ as $- 1 (- 8)$ .

$y = \frac{6 (1 + x)}{- (9 x) - 1 (- 8)}$

Step 3.4.4.3.5

Factor $- 1$ out of $- (9 x) - 1 (- 8)$ .

$y = \frac{6 (1 + x)}{- (9 x - 8)}$

Step 3.4.4.3.6

Rewrite negatives.

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Step 3.4.4.3.6.1

Rewrite $- (9 x - 8)$ as $- 1 (9 x - 8)$ .

$y = \frac{6 (1 + x)}{- 1 (9 x - 8)}$

Step 3.4.4.3.6.2

Move the negative in front of the fraction.

$y = - \frac{6 (1 + x)}{9 x - 8}$

Step 4

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

$f^{- 1} (x) = - \frac{6 (1 + x)}{9 x - 8}$

Step 5

Verify if $f^{- 1} (x) = - \frac{6 (1 + x)}{9 x - 8}$ is the inverse of $f (x) = \frac{- 8 x + 6}{- 9 x - 6}$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{6 (1 + \frac{- 8 x + 6}{- 9 x - 6})}{9 (\frac{- 8 x + 6}{- 9 x - 6}) - 8}$

Step 5.2.3

Simplify with factoring out.

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Step 5.2.3.1

Remove parentheses.

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{6 (1 + \frac{- 8 x + 6}{- 9 x - 6})}{9 (\frac{- 8 x + 6}{- 9 x - 6}) - 8}$

Step 5.2.3.2

Factor $2$ out of $- 8 x + 6$ .

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Step 5.2.3.2.1

Factor $2$ out of $- 8 x$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{6 (1 + \frac{2 (- 4 x) + 6}{- 9 x - 6})}{9 (\frac{- 8 x + 6}{- 9 x - 6}) - 8}$

Step 5.2.3.2.2

Factor $2$ out of $6$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{6 (1 + \frac{2 (- 4 x) + 2 (3)}{- 9 x - 6})}{9 (\frac{- 8 x + 6}{- 9 x - 6}) - 8}$

Step 5.2.3.2.3

Factor $2$ out of $2 (- 4 x) + 2 (3)$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{6 (1 + \frac{2 (- 4 x + 3)}{- 9 x - 6})}{9 (\frac{- 8 x + 6}{- 9 x - 6}) - 8}$

Step 5.2.3.3

Factor $3$ out of $- 9 x - 6$ .

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Step 5.2.3.3.1

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

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{6 (1 + \frac{2 (- 4 x + 3)}{3 (- 3 x) - 6})}{9 (\frac{- 8 x + 6}{- 9 x - 6}) - 8}$

Step 5.2.3.3.2

Factor $3$ out of $- 6$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{6 (1 + \frac{2 (- 4 x + 3)}{3 (- 3 x) + 3 (- 2)})}{9 (\frac{- 8 x + 6}{- 9 x - 6}) - 8}$

Step 5.2.3.3.3

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

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{6 (1 + \frac{2 (- 4 x + 3)}{3 (- 3 x - 2)})}{9 (\frac{- 8 x + 6}{- 9 x - 6}) - 8}$

Step 5.2.3.4

Factor $2$ out of $- 8 x + 6$ .

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Step 5.2.3.4.1

Factor $2$ out of $- 8 x$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{6 (1 + \frac{2 (- 4 x + 3)}{3 (- 3 x - 2)})}{9 (\frac{2 (- 4 x) + 6}{- 9 x - 6}) - 8}$

Step 5.2.3.4.2

Factor $2$ out of $6$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{6 (1 + \frac{2 (- 4 x + 3)}{3 (- 3 x - 2)})}{9 (\frac{2 (- 4 x) + 2 (3)}{- 9 x - 6}) - 8}$

Step 5.2.3.4.3

Factor $2$ out of $2 (- 4 x) + 2 (3)$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{6 (1 + \frac{2 (- 4 x + 3)}{3 (- 3 x - 2)})}{9 (\frac{2 (- 4 x + 3)}{- 9 x - 6}) - 8}$

Step 5.2.3.5

Factor $3$ out of $- 9 x - 6$ .

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Step 5.2.3.5.1

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

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{6 (1 + \frac{2 (- 4 x + 3)}{3 (- 3 x - 2)})}{9 (\frac{2 (- 4 x + 3)}{3 (- 3 x) - 6}) - 8}$

Step 5.2.3.5.2

Factor $3$ out of $- 6$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{6 (1 + \frac{2 (- 4 x + 3)}{3 (- 3 x - 2)})}{9 (\frac{2 (- 4 x + 3)}{3 (- 3 x) + 3 (- 2)}) - 8}$

Step 5.2.3.5.3

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

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{6 (1 + \frac{2 (- 4 x + 3)}{3 (- 3 x - 2)})}{9 (\frac{2 (- 4 x + 3)}{3 (- 3 x - 2)}) - 8}$

Step 5.2.4

Simplify the numerator.

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Step 5.2.4.1

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

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{6 (\frac{3 (- 3 x - 2)}{3 (- 3 x - 2)} + \frac{2 (- 4 x + 3)}{3 (- 3 x - 2)})}{9 (\frac{2 (- 4 x + 3)}{3 (- 3 x - 2)}) - 8}$

Step 5.2.4.2

Combine the numerators over the common denominator.

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{6 (\frac{3 (- 3 x - 2) + 2 (- 4 x + 3)}{3 (- 3 x - 2)})}{9 (\frac{2 (- 4 x + 3)}{3 (- 3 x - 2)}) - 8}$

Step 5.2.4.3

Rewrite $\frac{3 (- 3 x - 2) + 2 (- 4 x + 3)}{3 (- 3 x - 2)}$ in a factored form.

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Step 5.2.4.3.1

Apply the distributive property.

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{6 (\frac{3 (- 3 x) + 3 \cdot - 2 + 2 (- 4 x + 3)}{3 (- 3 x - 2)})}{9 (\frac{2 (- 4 x + 3)}{3 (- 3 x - 2)}) - 8}$

Step 5.2.4.3.2

Multiply $- 3$ by $3$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{6 (\frac{- 9 x + 3 \cdot - 2 + 2 (- 4 x + 3)}{3 (- 3 x - 2)})}{9 (\frac{2 (- 4 x + 3)}{3 (- 3 x - 2)}) - 8}$

Step 5.2.4.3.3

Multiply $3$ by $- 2$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{6 (\frac{- 9 x - 6 + 2 (- 4 x + 3)}{3 (- 3 x - 2)})}{9 (\frac{2 (- 4 x + 3)}{3 (- 3 x - 2)}) - 8}$

Step 5.2.4.3.4

Apply the distributive property.

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{6 (\frac{- 9 x - 6 + 2 (- 4 x) + 2 \cdot 3}{3 (- 3 x - 2)})}{9 (\frac{2 (- 4 x + 3)}{3 (- 3 x - 2)}) - 8}$

Step 5.2.4.3.5

Multiply $- 4$ by $2$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{6 (\frac{- 9 x - 6 - 8 x + 2 \cdot 3}{3 (- 3 x - 2)})}{9 (\frac{2 (- 4 x + 3)}{3 (- 3 x - 2)}) - 8}$

Step 5.2.4.3.6

Multiply $2$ by $3$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{6 (\frac{- 9 x - 6 - 8 x + 6}{3 (- 3 x - 2)})}{9 (\frac{2 (- 4 x + 3)}{3 (- 3 x - 2)}) - 8}$

Step 5.2.4.3.7

Subtract $8 x$ from $- 9 x$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{6 (\frac{- 17 x - 6 + 6}{3 (- 3 x - 2)})}{9 (\frac{2 (- 4 x + 3)}{3 (- 3 x - 2)}) - 8}$

Step 5.2.4.3.8

Add $- 6$ and $6$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{6 (\frac{- 17 x + 0}{3 (- 3 x - 2)})}{9 (\frac{2 (- 4 x + 3)}{3 (- 3 x - 2)}) - 8}$

Step 5.2.4.3.9

Add $- 17 x$ and $0$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{6 (\frac{- 17 x}{3 (- 3 x - 2)})}{9 (\frac{2 (- 4 x + 3)}{3 (- 3 x - 2)}) - 8}$

Step 5.2.4.4

Move the negative in front of the fraction.

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{6 (- \frac{(17) x}{3 (- 3 x - 2)})}{9 (\frac{2 (- 4 x + 3)}{3 (- 3 x - 2)}) - 8}$

Step 5.2.4.5

Remove unnecessary parentheses.

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{6 \cdot (- 1 \frac{17 x}{3 (- 3 x - 2)})}{9 (\frac{2 (- 4 x + 3)}{3 (- 3 x - 2)}) - 8}$

Step 5.2.4.6

Combine exponents.

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Step 5.2.4.6.1

Factor out negative.

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{- (6 (\frac{17 x}{3 (- 3 x - 2)}))}{9 (\frac{2 (- 4 x + 3)}{3 (- 3 x - 2)}) - 8}$

Step 5.2.4.6.2

Combine $6$ and $\frac{17 x}{3 (- 3 x - 2)}$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{- \frac{6 (17 x)}{3 (- 3 x - 2)}}{9 (\frac{2 (- 4 x + 3)}{3 (- 3 x - 2)}) - 8}$

Step 5.2.4.6.3

Multiply $17$ by $6$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{- \frac{102 x}{3 (- 3 x - 2)}}{9 (\frac{2 (- 4 x + 3)}{3 (- 3 x - 2)}) - 8}$

Step 5.2.4.7

Reduce the expression $\frac{102 x}{3 (- 3 x - 2)}$ by cancelling the common factors.

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Step 5.2.4.7.1

Factor $3$ out of $102 x$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{- \frac{3 (34 x)}{3 (- 3 x - 2)}}{9 (\frac{2 (- 4 x + 3)}{3 (- 3 x - 2)}) - 8}$

Step 5.2.4.7.2

Cancel the common factor.

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{- \frac{3 (34 x)}{3 (- 3 x - 2)}}{9 (\frac{2 (- 4 x + 3)}{3 (- 3 x - 2)}) - 8}$

Step 5.2.4.7.3

Rewrite the expression.

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{- \frac{34 x}{- 3 x - 2}}{9 (\frac{2 (- 4 x + 3)}{3 (- 3 x - 2)}) - 8}$

Step 5.2.5

Simplify the denominator.

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Step 5.2.5.1

Cancel the common factor of $3$ .

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Step 5.2.5.1.1

Factor $3$ out of $9$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{- \frac{34 x}{- 3 x - 2}}{3 (3) (\frac{2 (- 4 x + 3)}{3 (- 3 x - 2)}) - 8}$

Step 5.2.5.1.2

Cancel the common factor.

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{- \frac{34 x}{- 3 x - 2}}{3 \cdot (3 (\frac{2 (- 4 x + 3)}{3 (- 3 x - 2)})) - 8}$

Step 5.2.5.1.3

Rewrite the expression.

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{- \frac{34 x}{- 3 x - 2}}{3 (\frac{2 (- 4 x + 3)}{- 3 x - 2}) - 8}$

Step 5.2.5.2

Combine $3$ and $\frac{2 (- 4 x + 3)}{- 3 x - 2}$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{- \frac{34 x}{- 3 x - 2}}{\frac{3 (2 (- 4 x + 3))}{- 3 x - 2} - 8}$

Step 5.2.5.3

Multiply $2$ by $3$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{- \frac{34 x}{- 3 x - 2}}{\frac{6 (- 4 x + 3)}{- 3 x - 2} - 8}$

Step 5.2.5.4

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{- 3 x - 2}{- 3 x - 2}$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{- \frac{34 x}{- 3 x - 2}}{\frac{6 (- 4 x + 3)}{- 3 x - 2} - 8 \cdot \frac{- 3 x - 2}{- 3 x - 2}}$

Step 5.2.5.5

Combine $- 8$ and $\frac{- 3 x - 2}{- 3 x - 2}$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{- \frac{34 x}{- 3 x - 2}}{\frac{6 (- 4 x + 3)}{- 3 x - 2} + \frac{- 8 (- 3 x - 2)}{- 3 x - 2}}$

Step 5.2.5.6

Combine the numerators over the common denominator.

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{- \frac{34 x}{- 3 x - 2}}{\frac{6 (- 4 x + 3) - 8 (- 3 x - 2)}{- 3 x - 2}}$

Step 5.2.5.7

Rewrite $\frac{6 (- 4 x + 3) - 8 (- 3 x - 2)}{- 3 x - 2}$ in a factored form.

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Step 5.2.5.7.1

Factor $2$ out of $6 (- 4 x + 3) - 8 (- 3 x - 2)$ .

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Step 5.2.5.7.1.1

Factor $2$ out of $6 (- 4 x + 3)$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{- \frac{34 x}{- 3 x - 2}}{\frac{2 (3 (- 4 x + 3)) - 8 (- 3 x - 2)}{- 3 x - 2}}$

Step 5.2.5.7.1.2

Factor $2$ out of $- 8 (- 3 x - 2)$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{- \frac{34 x}{- 3 x - 2}}{\frac{2 (3 (- 4 x + 3)) + 2 (- 4 (- 3 x - 2))}{- 3 x - 2}}$

Step 5.2.5.7.1.3

Factor $2$ out of $2 (3 (- 4 x + 3)) + 2 (- 4 (- 3 x - 2))$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{- \frac{34 x}{- 3 x - 2}}{\frac{2 (3 (- 4 x + 3) - 4 (- 3 x - 2))}{- 3 x - 2}}$

Step 5.2.5.7.2

Apply the distributive property.

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{- \frac{34 x}{- 3 x - 2}}{\frac{2 (3 (- 4 x) + 3 \cdot 3 - 4 (- 3 x - 2))}{- 3 x - 2}}$

Step 5.2.5.7.3

Multiply $- 4$ by $3$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{- \frac{34 x}{- 3 x - 2}}{\frac{2 (- 12 x + 3 \cdot 3 - 4 (- 3 x - 2))}{- 3 x - 2}}$

Step 5.2.5.7.4

Multiply $3$ by $3$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{- \frac{34 x}{- 3 x - 2}}{\frac{2 (- 12 x + 9 - 4 (- 3 x - 2))}{- 3 x - 2}}$

Step 5.2.5.7.5

Apply the distributive property.

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{- \frac{34 x}{- 3 x - 2}}{\frac{2 (- 12 x + 9 - 4 (- 3 x) - 4 \cdot - 2)}{- 3 x - 2}}$

Step 5.2.5.7.6

Multiply $- 3$ by $- 4$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{- \frac{34 x}{- 3 x - 2}}{\frac{2 (- 12 x + 9 + 12 x - 4 \cdot - 2)}{- 3 x - 2}}$

Step 5.2.5.7.7

Multiply $- 4$ by $- 2$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{- \frac{34 x}{- 3 x - 2}}{\frac{2 (- 12 x + 9 + 12 x + 8)}{- 3 x - 2}}$

Step 5.2.5.7.8

Add $- 12 x$ and $12 x$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{- \frac{34 x}{- 3 x - 2}}{\frac{2 (0 + 9 + 8)}{- 3 x - 2}}$

Step 5.2.5.7.9

Add $0$ and $9$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{- \frac{34 x}{- 3 x - 2}}{\frac{2 (9 + 8)}{- 3 x - 2}}$

Step 5.2.5.7.10

Add $9$ and $8$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{- \frac{34 x}{- 3 x - 2}}{\frac{2 \cdot 17}{- 3 x - 2}}$

Step 5.2.5.8

Multiply $2$ by $17$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{- \frac{34 x}{- 3 x - 2}}{\frac{34}{- 3 x - 2}}$

Step 5.2.6

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - (- \frac{34 x}{- 3 x - 2} \cdot \frac{- 3 x - 2}{34})$

Step 5.2.7

Cancel the common factor of $34$ .

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Step 5.2.7.1

Move the leading negative in $- \frac{34 x}{- 3 x - 2}$ into the numerator.

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - (\frac{- 34 x}{- 3 x - 2} \cdot \frac{- 3 x - 2}{34})$

Step 5.2.7.2

Factor $34$ out of $- 34 x$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - (\frac{34 (- x)}{- 3 x - 2} \cdot \frac{- 3 x - 2}{34})$

Step 5.2.7.3

Cancel the common factor.

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - (\frac{34 (- x)}{- 3 x - 2} \cdot \frac{- 3 x - 2}{34})$

Step 5.2.7.4

Rewrite the expression.

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - (\frac{- x}{- 3 x - 2} \cdot (- 3 x - 2))$

Step 5.2.8

Move the negative in front of the fraction.

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - (- \frac{x}{- 3 x - 2} \cdot (- 3 x - 2))$

Step 5.2.9

Apply the distributive property.

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - (- \frac{x}{- 3 x - 2} \cdot (- 3 x) - \frac{x}{- 3 x - 2} \cdot - 2)$

Step 5.2.10

Multiply $- \frac{x}{- 3 x - 2} (- 3 x)$ .

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Step 5.2.10.1

Multiply $- 3$ by $- 1$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - (3 (\frac{x}{- 3 x - 2}) \cdot x - \frac{x}{- 3 x - 2} \cdot - 2)$

Step 5.2.10.2

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

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - (\frac{3 x}{- 3 x - 2} \cdot x - \frac{x}{- 3 x - 2} \cdot - 2)$

Step 5.2.10.3

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

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - (\frac{3 x \cdot x}{- 3 x - 2} - \frac{x}{- 3 x - 2} \cdot - 2)$

Step 5.2.10.4

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

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - (\frac{3 (x \cdot x)}{- 3 x - 2} - \frac{x}{- 3 x - 2} \cdot - 2)$

Step 5.2.10.5

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

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - (\frac{3 (x \cdot x)}{- 3 x - 2} - \frac{x}{- 3 x - 2} \cdot - 2)$

Step 5.2.10.6

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

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - (\frac{3 x^{1 + 1}}{- 3 x - 2} - \frac{x}{- 3 x - 2} \cdot - 2)$

Step 5.2.10.7

Add $1$ and $1$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - (\frac{3 x^{2}}{- 3 x - 2} - \frac{x}{- 3 x - 2} \cdot - 2)$

Step 5.2.11

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

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Step 5.2.11.1

Multiply $- 2$ by $- 1$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - (\frac{3 x^{2}}{- 3 x - 2} + 2 (\frac{x}{- 3 x - 2}))$

Step 5.2.11.2

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

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - (\frac{3 x^{2}}{- 3 x - 2} + \frac{2 x}{- 3 x - 2})$

Step 5.2.12

Combine the numerators over the common denominator.

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{3 x^{2} + 2 x}{- 3 x - 2}$

Step 5.2.13

Factor $x$ out of $3 x^{2} + 2 x$ .

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Step 5.2.13.1

Factor $x$ out of $3 x^{2}$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{x (3 x) + 2 x}{- 3 x - 2}$

Step 5.2.13.2

Factor $x$ out of $2 x$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{x (3 x) + x \cdot 2}{- 3 x - 2}$

Step 5.2.13.3

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

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{x (3 x + 2)}{- 3 x - 2}$

Step 5.2.14

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

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Step 5.2.14.1

Factor $- 1$ out of $3 x$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{x (- (- 3 x) + 2)}{- 3 x - 2}$

Step 5.2.14.2

Rewrite $2$ as $- 1 (- 2)$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{x (- (- 3 x) - 1 \cdot - 2)}{- 3 x - 2}$

Step 5.2.14.3

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

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{x (- (- 3 x - 2))}{- 3 x - 2}$

Step 5.2.14.4

Rewrite $- (- 3 x - 2)$ as $- 1 (- 3 x - 2)$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{x (- 1 (- 3 x - 2))}{- 3 x - 2}$

Step 5.2.14.5

Cancel the common factor.

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - \frac{x (- 1 (- 3 x - 2))}{- 3 x - 2}$

Step 5.2.14.6

Divide $x \cdot (- 1)$ by $1$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - (x \cdot (- 1))$

Step 5.2.15

Simplify the expression.

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Step 5.2.15.1

Move $- 1$ to the left of $x$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = - (- 1 x)$

Step 5.2.15.2

Rewrite $- 1 x$ as $- x$ .

$f^{- 1} (\frac{- 8 x + 6}{- 9 x - 6}) = x$

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $f (- \frac{6 (1 + x)}{9 x - 8})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{- 8 (- \frac{6 (1 + x)}{9 x - 8}) + 6}{- 9 (- \frac{6 (1 + x)}{9 x - 8}) - 6}$

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Factor $2$ out of $- 8 (- \frac{6 (1 + x)}{9 x - 8}) + 6$ .

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Step 5.3.3.1.1

Factor $2$ out of $- 8 (- \frac{6 (1 + x)}{9 x - 8})$ .

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (- 4 (- \frac{6 (1 + x)}{9 x - 8})) + 6}{- 9 (- \frac{6 (1 + x)}{9 x - 8}) - 6}$

Step 5.3.3.1.2

Factor $2$ out of $6$ .

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (- 4 (- \frac{6 (1 + x)}{9 x - 8})) + 2 (3)}{- 9 (- \frac{6 (1 + x)}{9 x - 8}) - 6}$

Step 5.3.3.1.3

Factor $2$ out of $2 (- 4 (- \frac{6 (1 + x)}{9 x - 8})) + 2 (3)$ .

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (- 4 (- \frac{6 (1 + x)}{9 x - 8}) + 3)}{- 9 (- \frac{6 (1 + x)}{9 x - 8}) - 6}$

Step 5.3.3.2

Multiply $- 4 (- \frac{6 (1 + x)}{9 x - 8})$ .

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Step 5.3.3.2.1

Multiply $- 1$ by $- 4$ .

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (4 (\frac{6 (1 + x)}{9 x - 8}) + 3)}{- 9 (- \frac{6 (1 + x)}{9 x - 8}) - 6}$

Step 5.3.3.2.2

Combine $4$ and $\frac{6 (1 + x)}{9 x - 8}$ .

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (\frac{4 (6 (1 + x))}{9 x - 8} + 3)}{- 9 (- \frac{6 (1 + x)}{9 x - 8}) - 6}$

Step 5.3.3.2.3

Multiply $6$ by $4$ .

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (\frac{24 (1 + x)}{9 x - 8} + 3)}{- 9 (- \frac{6 (1 + x)}{9 x - 8}) - 6}$

Step 5.3.3.3

To write $3$ as a fraction with a common denominator, multiply by $\frac{9 x - 8}{9 x - 8}$ .

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (\frac{24 (1 + x)}{9 x - 8} + \frac{3 (9 x - 8)}{9 x - 8})}{- 9 (- \frac{6 (1 + x)}{9 x - 8}) - 6}$

Step 5.3.3.4

Combine the numerators over the common denominator.

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (\frac{24 (1 + x) + 3 (9 x - 8)}{9 x - 8})}{- 9 (- \frac{6 (1 + x)}{9 x - 8}) - 6}$

Step 5.3.3.5

Rewrite $\frac{24 (1 + x) + 3 (9 x - 8)}{9 x - 8}$ in a factored form.

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Step 5.3.3.5.1

Factor $3$ out of $24 (1 + x) + 3 (9 x - 8)$ .

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Step 5.3.3.5.1.1

Factor $3$ out of $24 (1 + x)$ .

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (\frac{3 (8 (1 + x)) + 3 (9 x - 8)}{9 x - 8})}{- 9 (- \frac{6 (1 + x)}{9 x - 8}) - 6}$

Step 5.3.3.5.1.2

Factor $3$ out of $3 (8 (1 + x)) + 3 (9 x - 8)$ .

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (\frac{3 (8 (1 + x) + 9 x - 8)}{9 x - 8})}{- 9 (- \frac{6 (1 + x)}{9 x - 8}) - 6}$

Step 5.3.3.5.2

Apply the distributive property.

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (\frac{3 (8 \cdot 1 + 8 x + 9 x - 8)}{9 x - 8})}{- 9 (- \frac{6 (1 + x)}{9 x - 8}) - 6}$

Step 5.3.3.5.3

Multiply $8$ by $1$ .

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (\frac{3 (8 + 8 x + 9 x - 8)}{9 x - 8})}{- 9 (- \frac{6 (1 + x)}{9 x - 8}) - 6}$

Step 5.3.3.5.4

Subtract $8$ from $8$ .

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (\frac{3 (8 x + 9 x + 0)}{9 x - 8})}{- 9 (- \frac{6 (1 + x)}{9 x - 8}) - 6}$

Step 5.3.3.5.5

Add $8 x + 9 x$ and $0$ .

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (\frac{3 (8 x + 9 x)}{9 x - 8})}{- 9 (- \frac{6 (1 + x)}{9 x - 8}) - 6}$

Step 5.3.3.5.6

Add $8 x$ and $9 x$ .

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (\frac{3 \cdot (17 x)}{9 x - 8})}{- 9 (- \frac{6 (1 + x)}{9 x - 8}) - 6}$

Step 5.3.3.5.7

Multiply $3$ by $17$ .

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (\frac{51 x}{9 x - 8})}{- 9 (- \frac{6 (1 + x)}{9 x - 8}) - 6}$

Step 5.3.4

Simplify the denominator.

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Step 5.3.4.1

Factor $- 3$ out of $- 9 (- \frac{6 (1 + x)}{9 x - 8}) - 6$ .

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Step 5.3.4.1.1

Reorder $1$ and $x$ .

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (\frac{51 x}{9 x - 8})}{- 9 \cdot (- 1 \frac{6 (x + 1)}{9 x - 8}) - 6}$

Step 5.3.4.1.2

Factor $- 3$ out of $- 9 \cdot - 1 \frac{6 (x + 1)}{9 x - 8}$ .

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (\frac{51 x}{9 x - 8})}{- 3 (3 \cdot (- 1 \frac{6 (x + 1)}{9 x - 8})) - 6}$

Step 5.3.4.1.3

Factor $- 3$ out of $- 6$ .

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (\frac{51 x}{9 x - 8})}{- 3 (3 \cdot (- 1 \frac{6 (x + 1)}{9 x - 8})) - 3 \cdot 2}$

Step 5.3.4.1.4

Factor $- 3$ out of $- 3 (3 \cdot - 1 \frac{6 (x + 1)}{9 x - 8}) - 3 (2)$ .

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (\frac{51 x}{9 x - 8})}{- 3 (3 \cdot (- 1 \frac{6 (x + 1)}{9 x - 8}) + 2)}$

Step 5.3.4.2

Multiply $3$ by $- 1$ .

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (\frac{51 x}{9 x - 8})}{- 3 (- 3 \frac{6 (x + 1)}{9 x - 8} + 2)}$

Step 5.3.4.3

Multiply $- 3 \frac{6 (x + 1)}{9 x - 8}$ .

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Step 5.3.4.3.1

Combine $- 3$ and $\frac{6 (x + 1)}{9 x - 8}$ .

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (\frac{51 x}{9 x - 8})}{- 3 (\frac{- 3 (6 (x + 1))}{9 x - 8} + 2)}$

Step 5.3.4.3.2

Multiply $6$ by $- 3$ .

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (\frac{51 x}{9 x - 8})}{- 3 (\frac{- 18 (x + 1)}{9 x - 8} + 2)}$

Step 5.3.4.4

Move the negative in front of the fraction.

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (\frac{51 x}{9 x - 8})}{- 3 (- \frac{(18) (x + 1)}{9 x - 8} + 2)}$

Step 5.3.4.5

To write $2$ as a fraction with a common denominator, multiply by $\frac{9 x - 8}{9 x - 8}$ .

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (\frac{51 x}{9 x - 8})}{- 3 (- \frac{18 (x + 1)}{9 x - 8} + \frac{2 (9 x - 8)}{9 x - 8})}$

Step 5.3.4.6

Combine the numerators over the common denominator.

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (\frac{51 x}{9 x - 8})}{- 3 \frac{- 18 (x + 1) + 2 (9 x - 8)}{9 x - 8}}$

Step 5.3.4.7

Reorder terms.

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (\frac{51 x}{9 x - 8})}{- 3 \frac{2 (9 x - 8) - 18 (x + 1)}{9 x - 8}}$

Step 5.3.4.8

Rewrite $\frac{2 (9 x - 8) - 18 (x + 1)}{9 x - 8}$ in a factored form.

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Step 5.3.4.8.1

Factor $2$ out of $2 (9 x - 8) - 18 (x + 1)$ .

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Step 5.3.4.8.1.1

Factor $2$ out of $- 18 (x + 1)$ .

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (\frac{51 x}{9 x - 8})}{- 3 \frac{2 (9 x - 8) + 2 (- 9 (x + 1))}{9 x - 8}}$

Step 5.3.4.8.1.2

Factor $2$ out of $2 (9 x - 8) + 2 (- 9 (x + 1))$ .

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (\frac{51 x}{9 x - 8})}{- 3 \frac{2 (9 x - 8 - 9 (x + 1))}{9 x - 8}}$

Step 5.3.4.8.2

Apply the distributive property.

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (\frac{51 x}{9 x - 8})}{- 3 \frac{2 (9 x - 8 - 9 x - 9 \cdot 1)}{9 x - 8}}$

Step 5.3.4.8.3

Multiply $- 9$ by $1$ .

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (\frac{51 x}{9 x - 8})}{- 3 \frac{2 (9 x - 8 - 9 x - 9)}{9 x - 8}}$

Step 5.3.4.8.4

Subtract $9 x$ from $9 x$ .

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (\frac{51 x}{9 x - 8})}{- 3 \frac{2 (0 - 8 - 9)}{9 x - 8}}$

Step 5.3.4.8.5

Subtract $8$ from $0$ .

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (\frac{51 x}{9 x - 8})}{- 3 \frac{2 (- 8 - 9)}{9 x - 8}}$

Step 5.3.4.8.6

Subtract $9$ from $- 8$ .

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (\frac{51 x}{9 x - 8})}{- 3 \frac{2 \cdot - 17}{9 x - 8}}$

Step 5.3.4.9

Multiply $2$ by $- 17$ .

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (\frac{51 x}{9 x - 8})}{- 3 \frac{- 34}{9 x - 8}}$

Step 5.3.4.10

Move the negative in front of the fraction.

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (\frac{51 x}{9 x - 8})}{- 3 \cdot (- 1 \frac{34}{9 x - 8})}$

Step 5.3.4.11

Combine exponents.

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Step 5.3.4.11.1

Factor out negative.

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (\frac{51 x}{9 x - 8})}{- (- 3 \frac{34}{9 x - 8})}$

Step 5.3.4.11.2

Combine $- 3$ and $\frac{34}{9 x - 8}$ .

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (\frac{51 x}{9 x - 8})}{- \frac{- 3 \cdot 34}{9 x - 8}}$

Step 5.3.4.11.3

Multiply $- 3$ by $34$ .

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (\frac{51 x}{9 x - 8})}{- \frac{- 102}{9 x - 8}}$

Step 5.3.4.12

Move the negative in front of the fraction.

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{2 (\frac{51 x}{9 x - 8})}{\frac{102}{9 x - 8}}$

Step 5.3.5

Combine $2$ and $\frac{51 x}{9 x - 8}$ .

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{\frac{2 (51 x)}{9 x - 8}}{\frac{102}{9 x - 8}}$

Step 5.3.6

Multiply $2$ by $51$ .

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{\frac{102 x}{9 x - 8}}{\frac{102}{9 x - 8}}$

Step 5.3.7

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{102 x}{9 x - 8} \cdot \frac{9 x - 8}{102}$

Step 5.3.8

Cancel the common factor of $102$ .

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Step 5.3.8.1

Factor $102$ out of $102 x$ .

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{102 (x)}{9 x - 8} \cdot \frac{9 x - 8}{102}$

Step 5.3.8.2

Cancel the common factor.

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{102 x}{9 x - 8} \cdot \frac{9 x - 8}{102}$

Step 5.3.8.3

Rewrite the expression.

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{x}{9 x - 8} \cdot (9 x - 8)$

Step 5.3.9

Cancel the common factor of $9 x - 8$ .

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Step 5.3.9.1

Cancel the common factor.

$f (- \frac{6 (1 + x)}{9 x - 8}) = \frac{x}{9 x - 8} \cdot (9 x - 8)$

Step 5.3.9.2

Rewrite the expression.

$f (- \frac{6 (1 + x)}{9 x - 8}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{6 (1 + x)}{9 x - 8}$ is the inverse of $f (x) = \frac{- 8 x + 6}{- 9 x - 6}$ .

$f^{- 1} (x) = - \frac{6 (1 + x)}{9 x - 8}$