Finite Math Examples

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

Step 1

Write $f (x) = \frac{x - 1}{x - 6}$ as an equation.

$y = \frac{x - 1}{x - 6}$

Step 2

Interchange the variables.

$x = \frac{y - 1}{y - 6}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{y - 1}{y - 6} = x$ .

$\frac{y - 1}{y - 6} = x$

Step 3.2

Find the LCD of the terms in the equation.

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Step 3.2.1

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

$y - 6, 1$

Step 3.2.2

Remove parentheses.

$y - 6, 1$

Step 3.2.3

The LCM of one and any expression is the expression.

$y - 6$

Step 3.3

Multiply each term in $\frac{y - 1}{y - 6} = x$ by $y - 6$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $\frac{y - 1}{y - 6} = x$ by $y - 6$ .

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $y - 6$ .

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Step 3.3.2.1.1

Cancel the common factor.

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

Step 3.3.2.1.2

Rewrite the expression.

$y - 1 = x (y - 6)$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Apply the distributive property.

$y - 1 = x y + x \cdot - 6$

Step 3.3.3.2

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

$y - 1 = x y - 6 x$

Step 3.4

Solve the equation.

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Step 3.4.1

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

$y - 1 - x y = - 6 x$

Step 3.4.2

Add $1$ to both sides of the equation.

$y - x y = - 6 x + 1$

Step 3.4.3

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

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Step 3.4.3.1

Factor $y$ out of $y^{1}$ .

$y \cdot 1 - x y = - 6 x + 1$

Step 3.4.3.2

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

$y \cdot 1 + y (- x) = - 6 x + 1$

Step 3.4.3.3

Factor $y$ out of $y \cdot 1 + y (- x)$ .

$y (1 - x) = - 6 x + 1$

Step 3.4.4

Divide each term in $y (1 - x) = - 6 x + 1$ by $1 - x$ and simplify.

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Step 3.4.4.1

Divide each term in $y (1 - x) = - 6 x + 1$ by $1 - x$ .

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

Step 3.4.4.2

Simplify the left side.

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Step 3.4.4.2.1

Cancel the common factor of $1 - x$ .

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Step 3.4.4.2.1.1

Cancel the common factor.

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

Step 3.4.4.2.1.2

Divide $y$ by $1$ .

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

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.

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

Step 3.4.4.3.2

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

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

Step 3.4.4.3.3

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

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

Step 3.4.4.3.4

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

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

Step 3.4.4.3.5

Simplify the expression.

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Step 3.4.4.3.5.1

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

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

Step 3.4.4.3.5.2

Move the negative in front of the fraction.

$y = - \frac{6 x - 1}{1 - x}$

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Multiply the numerator and denominator of the fraction by $x - 6$ .

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Step 5.2.3.1

Multiply $\frac{6 \frac{x - 1}{x - 6} - 1}{1 - \frac{x - 1}{x - 6}}$ by $\frac{x - 6}{x - 6}$ .

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

Step 5.2.3.2

Combine.

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

Step 5.2.4

Apply the distributive property.

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

Step 5.2.5

Cancel the common factor of $x - 6$ .

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Step 5.2.5.1

Move the leading negative in $- \frac{x - 1}{x - 6}$ into the numerator.

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

Step 5.2.5.2

Cancel the common factor.

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

Step 5.2.5.3

Rewrite the expression.

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

Step 5.2.6

Simplify the numerator.

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Step 5.2.6.1

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

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Step 5.2.6.1.1

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

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

Step 5.2.6.1.2

Factor $x - 6$ out of $(x - 6) \cdot - 1$ .

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

Step 5.2.6.1.3

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

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

Step 5.2.6.2

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

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

Step 5.2.6.3

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

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

Step 5.2.6.4

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

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

Step 5.2.6.5

Combine the numerators over the common denominator.

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

Step 5.2.6.6

Rewrite $\frac{6 (x - 1) - (x - 6)}{x - 6}$ in a factored form.

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Step 5.2.6.6.1

Apply the distributive property.

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

Step 5.2.6.6.2

Multiply $6$ by $- 1$ .

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

Step 5.2.6.6.3

Apply the distributive property.

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

Step 5.2.6.6.4

Multiply $- 1$ by $- 6$ .

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

Step 5.2.6.6.5

Subtract $x$ from $6 x$ .

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

Step 5.2.6.6.6

Add $- 6$ and $6$ .

$f^{- 1} (\frac{x - 1}{x - 6}) = - \frac{(x - 6) (\frac{5 x + 0}{x - 6})}{(x - 6) \cdot 1 - (x - 1)}$

Step 5.2.6.6.7

Add $5 x$ and $0$ .

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

Step 5.2.7

Simplify the denominator.

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Step 5.2.7.1

Multiply $x - 6$ by $1$ .

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

Step 5.2.7.2

Apply the distributive property.

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

Step 5.2.7.3

Multiply $- 1$ by $- 1$ .

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

Step 5.2.7.4

Subtract $x$ from $x$ .

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

Step 5.2.7.5

Subtract $6$ from $0$ .

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

Step 5.2.7.6

Add $- 6$ and $1$ .

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

Step 5.2.8

Simplify terms.

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Step 5.2.8.1

Factor $\frac{5 x}{x - 6}$ out of $\frac{(x - 6) \frac{5 x}{x - 6}}{- 5}$ .

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

Step 5.2.8.2

Cancel the common factor of $5$ .

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Step 5.2.8.2.1

Factor $5$ out of $5 x$ .

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

Step 5.2.8.2.2

Factor $5$ out of $- 5$ .

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

Step 5.2.8.2.3

Cancel the common factor.

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

Step 5.2.8.2.4

Rewrite the expression.

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

Step 5.2.8.3

Cancel the common factor of $x - 6$ .

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Step 5.2.8.3.1

Cancel the common factor.

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

Step 5.2.8.3.2

Rewrite the expression.

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

Step 5.2.8.4

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

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

Step 5.2.8.5

Simplify the expression.

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Step 5.2.8.5.1

Move the negative one from the denominator of $\frac{x}{- 1}$ .

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

Step 5.2.8.5.2

Rewrite $- 1 \cdot x$ as $- x$ .

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

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

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

Step 5.3.3

Multiply the numerator and denominator of the fraction by $1 - x$ .

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Step 5.3.3.1

Multiply $\frac{- \frac{6 x - 1}{1 - x} - 1}{- \frac{6 x - 1}{1 - x} - 6}$ by $\frac{1 - x}{1 - x}$ .

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

Step 5.3.3.2

Combine.

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

Step 5.3.4

Apply the distributive property.

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

Step 5.3.5

Simplify by cancelling.

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Step 5.3.5.1

Cancel the common factor of $1 - x$ .

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Step 5.3.5.1.1

Move the leading negative in $- \frac{6 x - 1}{1 - x}$ into the numerator.

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

Step 5.3.5.1.2

Cancel the common factor.

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

Step 5.3.5.1.3

Rewrite the expression.

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

Step 5.3.5.2

Cancel the common factor of $1 - x$ .

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Step 5.3.5.2.1

Move the leading negative in $- \frac{6 x - 1}{1 - x}$ into the numerator.

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

Step 5.3.5.2.2

Cancel the common factor.

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

Step 5.3.5.2.3

Rewrite the expression.

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

Step 5.3.6

Simplify the numerator.

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Step 5.3.6.1

Apply the distributive property.

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

Step 5.3.6.2

Multiply $6$ by $- 1$ .

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

Step 5.3.6.3

Multiply $- 1$ by $- 1$ .

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

Step 5.3.6.4

Apply the distributive property.

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

Step 5.3.6.5

Multiply $- 1$ by $1$ .

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

Step 5.3.6.6

Multiply $- x \cdot - 1$ .

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Step 5.3.6.6.1

Multiply $- 1$ by $- 1$ .

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

Step 5.3.6.6.2

Multiply $x$ by $1$ .

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

Step 5.3.6.7

Add $- 6 x$ and $x$ .

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

Step 5.3.6.8

Subtract $1$ from $1$ .

$f (- \frac{6 x - 1}{1 - x}) = \frac{- 5 x + 0}{- (6 x - 1) + (1 - x) \cdot - 6}$

Step 5.3.6.9

Add $- 5 x$ and $0$ .

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

Step 5.3.7

Simplify the denominator.

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Step 5.3.7.1

Apply the distributive property.

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

Step 5.3.7.2

Multiply $6$ by $- 1$ .

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

Step 5.3.7.3

Multiply $- 1$ by $- 1$ .

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

Step 5.3.7.4

Apply the distributive property.

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

Step 5.3.7.5

Multiply $- 6$ by $1$ .

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

Step 5.3.7.6

Multiply $- 6$ by $- 1$ .

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

Step 5.3.7.7

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

$f (- \frac{6 x - 1}{1 - x}) = \frac{- 5 x}{0 + 1 - 6}$

Step 5.3.7.8

Add $0$ and $1$ .

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

Step 5.3.7.9

Subtract $6$ from $1$ .

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

Step 5.3.8

Cancel the common factor of $- 5$ .

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Step 5.3.8.1

Cancel the common factor.

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

Step 5.3.8.2

Divide $x$ by $1$ .

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

Step 5.4

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

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