Precalculus Examples

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

Step 1

Write $f (x) = \frac{6 x - 1}{5 x - 6}$ as an equation.

$y = \frac{6 x - 1}{5 x - 6}$

Step 2

Interchange the variables.

$x = \frac{6 y - 1}{5 y - 6}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{6 y - 1}{5 y - 6} = x$ .

$\frac{6 y - 1}{5 y - 6} = x$

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.

$5 y - 6, 1$

Step 3.2.2

Remove parentheses.

$5 y - 6, 1$

Step 3.2.3

The LCM of one and any expression is the expression.

$5 y - 6$

Step 3.3

Multiply each term in $\frac{6 y - 1}{5 y - 6} = x$ by $5 y - 6$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $\frac{6 y - 1}{5 y - 6} = x$ by $5 y - 6$ .

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

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

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Step 3.3.2.1.1

Cancel the common factor.

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

Step 3.3.2.1.2

Rewrite the expression.

$6 y - 1 = x (5 y - 6)$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Apply the distributive property.

$6 y - 1 = x (5 y) + x \cdot - 6$

Step 3.3.3.2

Reorder.

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Step 3.3.3.2.1

Rewrite using the commutative property of multiplication.

$6 y - 1 = 5 x y + x \cdot - 6$

Step 3.3.3.2.2

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

$6 y - 1 = 5 x y - 6 x$

Step 3.4

Solve the equation.

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Step 3.4.1

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

$6 y - 1 - 5 x y = - 6 x$

Step 3.4.2

Add $1$ to both sides of the equation.

$6 y - 5 x y = - 6 x + 1$

Step 3.4.3

Factor $y$ out of $6 y - 5 x y$ .

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Step 3.4.3.1

Factor $y$ out of $6 y$ .

$y \cdot 6 - 5 x y = - 6 x + 1$

Step 3.4.3.2

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

$y \cdot 6 + y (- 5 x) = - 6 x + 1$

Step 3.4.3.3

Factor $y$ out of $y \cdot 6 + y (- 5 x)$ .

$y (6 - 5 x) = - 6 x + 1$

Step 3.4.4

Divide each term in $y (6 - 5 x) = - 6 x + 1$ by $6 - 5 x$ and simplify.

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Step 3.4.4.1

Divide each term in $y (6 - 5 x) = - 6 x + 1$ by $6 - 5 x$ .

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

Step 3.4.4.2

Simplify the left side.

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Step 3.4.4.2.1

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

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Step 3.4.4.2.1.1

Cancel the common factor.

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

Step 3.4.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 6 x}{6 - 5 x} + \frac{1}{6 - 5 x}$

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.

$y = \frac{- 6 x + 1}{6 - 5 x}$

Step 3.4.4.3.2

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

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

Step 3.4.4.3.3

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

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

Step 3.4.4.3.4

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

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

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 $- 1 (6 x - 1)$ .

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

Step 3.4.4.3.5.2

Move the negative in front of the fraction.

$y = - \frac{6 x - 1}{6 - 5 x}$

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

Combine $6$ and $\frac{6 x - 1}{5 x - 6}$ .

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

Step 5.2.3.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5 x - 6}{5 x - 6}$ .

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

Step 5.2.3.3

Combine $- 1$ and $\frac{5 x - 6}{5 x - 6}$ .

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

Step 5.2.3.4

Combine the numerators over the common denominator.

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

Step 5.2.3.5

Rewrite $\frac{6 (6 x - 1) - (5 x - 6)}{5 x - 6}$ in a factored form.

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Step 5.2.3.5.1

Apply the distributive property.

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

Step 5.2.3.5.2

Multiply $6$ by $6$ .

$f^{- 1} (\frac{6 x - 1}{5 x - 6}) = - \frac{\frac{36 x + 6 \cdot - 1 - (5 x - 6)}{5 x - 6}}{6 - 5 \frac{6 x - 1}{5 x - 6}}$

Step 5.2.3.5.3

Multiply $6$ by $- 1$ .

$f^{- 1} (\frac{6 x - 1}{5 x - 6}) = - \frac{\frac{36 x - 6 - (5 x - 6)}{5 x - 6}}{6 - 5 \frac{6 x - 1}{5 x - 6}}$

Step 5.2.3.5.4

Apply the distributive property.

$f^{- 1} (\frac{6 x - 1}{5 x - 6}) = - \frac{\frac{36 x - 6 - (5 x) + 6}{5 x - 6}}{6 - 5 \frac{6 x - 1}{5 x - 6}}$

Step 5.2.3.5.5

Multiply $5$ by $- 1$ .

$f^{- 1} (\frac{6 x - 1}{5 x - 6}) = - \frac{\frac{36 x - 6 - 5 x + 6}{5 x - 6}}{6 - 5 \frac{6 x - 1}{5 x - 6}}$

Step 5.2.3.5.6

Multiply $- 1$ by $- 6$ .

$f^{- 1} (\frac{6 x - 1}{5 x - 6}) = - \frac{\frac{36 x - 6 - 5 x + 6}{5 x - 6}}{6 - 5 \frac{6 x - 1}{5 x - 6}}$

Step 5.2.3.5.7

Subtract $5 x$ from $36 x$ .

$f^{- 1} (\frac{6 x - 1}{5 x - 6}) = - \frac{\frac{31 x - 6 + 6}{5 x - 6}}{6 - 5 \frac{6 x - 1}{5 x - 6}}$

Step 5.2.3.5.8

Add $- 6$ and $6$ .

$f^{- 1} (\frac{6 x - 1}{5 x - 6}) = - \frac{\frac{31 x + 0}{5 x - 6}}{6 - 5 \frac{6 x - 1}{5 x - 6}}$

Step 5.2.3.5.9

Add $31 x$ and $0$ .

$f^{- 1} (\frac{6 x - 1}{5 x - 6}) = - \frac{\frac{31 x}{5 x - 6}}{6 - 5 \frac{6 x - 1}{5 x - 6}}$

Step 5.2.4

Simplify the denominator.

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Step 5.2.4.1

Combine $- 5$ and $\frac{6 x - 1}{5 x - 6}$ .

$f^{- 1} (\frac{6 x - 1}{5 x - 6}) = - \frac{\frac{31 x}{5 x - 6}}{6 + \frac{- 5 (6 x - 1)}{5 x - 6}}$

Step 5.2.4.2

Move the negative in front of the fraction.

$f^{- 1} (\frac{6 x - 1}{5 x - 6}) = - \frac{\frac{31 x}{5 x - 6}}{6 - \frac{(5) (6 x - 1)}{5 x - 6}}$

Step 5.2.4.3

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

$f^{- 1} (\frac{6 x - 1}{5 x - 6}) = - \frac{\frac{31 x}{5 x - 6}}{\frac{6 (5 x - 6)}{5 x - 6} - \frac{5 (6 x - 1)}{5 x - 6}}$

Step 5.2.4.4

Combine the numerators over the common denominator.

$f^{- 1} (\frac{6 x - 1}{5 x - 6}) = - \frac{\frac{31 x}{5 x - 6}}{\frac{6 (5 x - 6) - 5 (6 x - 1)}{5 x - 6}}$

Step 5.2.4.5

Rewrite $\frac{6 (5 x - 6) - 5 (6 x - 1)}{5 x - 6}$ in a factored form.

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Step 5.2.4.5.1

Apply the distributive property.

$f^{- 1} (\frac{6 x - 1}{5 x - 6}) = - \frac{\frac{31 x}{5 x - 6}}{\frac{6 (5 x) + 6 \cdot - 6 - 5 (6 x - 1)}{5 x - 6}}$

Step 5.2.4.5.2

Multiply $5$ by $6$ .

$f^{- 1} (\frac{6 x - 1}{5 x - 6}) = - \frac{\frac{31 x}{5 x - 6}}{\frac{30 x + 6 \cdot - 6 - 5 (6 x - 1)}{5 x - 6}}$

Step 5.2.4.5.3

Multiply $6$ by $- 6$ .

$f^{- 1} (\frac{6 x - 1}{5 x - 6}) = - \frac{\frac{31 x}{5 x - 6}}{\frac{30 x - 36 - 5 (6 x - 1)}{5 x - 6}}$

Step 5.2.4.5.4

Apply the distributive property.

$f^{- 1} (\frac{6 x - 1}{5 x - 6}) = - \frac{\frac{31 x}{5 x - 6}}{\frac{30 x - 36 - 5 (6 x) - 5 \cdot - 1}{5 x - 6}}$

Step 5.2.4.5.5

Multiply $6$ by $- 5$ .

$f^{- 1} (\frac{6 x - 1}{5 x - 6}) = - \frac{\frac{31 x}{5 x - 6}}{\frac{30 x - 36 - 30 x - 5 \cdot - 1}{5 x - 6}}$

Step 5.2.4.5.6

Multiply $- 5$ by $- 1$ .

$f^{- 1} (\frac{6 x - 1}{5 x - 6}) = - \frac{\frac{31 x}{5 x - 6}}{\frac{30 x - 36 - 30 x + 5}{5 x - 6}}$

Step 5.2.4.5.7

Subtract $30 x$ from $30 x$ .

$f^{- 1} (\frac{6 x - 1}{5 x - 6}) = - \frac{\frac{31 x}{5 x - 6}}{\frac{0 - 36 + 5}{5 x - 6}}$

Step 5.2.4.5.8

Subtract $36$ from $0$ .

$f^{- 1} (\frac{6 x - 1}{5 x - 6}) = - \frac{\frac{31 x}{5 x - 6}}{\frac{- 36 + 5}{5 x - 6}}$

Step 5.2.4.5.9

Add $- 36$ and $5$ .

$f^{- 1} (\frac{6 x - 1}{5 x - 6}) = - \frac{\frac{31 x}{5 x - 6}}{\frac{- 31}{5 x - 6}}$

Step 5.2.4.6

Move the negative in front of the fraction.

$f^{- 1} (\frac{6 x - 1}{5 x - 6}) = - \frac{\frac{31 x}{5 x - 6}}{- \frac{31}{5 x - 6}}$

Step 5.2.5

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{6 x - 1}{5 x - 6}) = - (\frac{31 x}{5 x - 6} \cdot (- \frac{5 x - 6}{31}))$

Step 5.2.6

Rewrite using the commutative property of multiplication.

$f^{- 1} (\frac{6 x - 1}{5 x - 6}) = - (- \frac{31 x}{5 x - 6} \cdot \frac{5 x - 6}{31})$

Step 5.2.7

Cancel the common factor of $31$ .

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Step 5.2.7.1

Move the leading negative in $- \frac{31 x}{5 x - 6}$ into the numerator.

$f^{- 1} (\frac{6 x - 1}{5 x - 6}) = - (\frac{- 31 x}{5 x - 6} \cdot \frac{5 x - 6}{31})$

Step 5.2.7.2

Factor $31$ out of $- 31 x$ .

$f^{- 1} (\frac{6 x - 1}{5 x - 6}) = - (\frac{31 (- x)}{5 x - 6} \cdot \frac{5 x - 6}{31})$

Step 5.2.7.3

Cancel the common factor.

$f^{- 1} (\frac{6 x - 1}{5 x - 6}) = - (\frac{31 (- x)}{5 x - 6} \cdot \frac{5 x - 6}{31})$

Step 5.2.7.4

Rewrite the expression.

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

Step 5.2.8

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

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Step 5.2.8.1

Cancel the common factor.

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

Step 5.2.8.2

Rewrite the expression.

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

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

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

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Multiply $6 (- \frac{6 x - 1}{6 - 5 x})$ .

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Step 5.3.3.1.1

Multiply $- 1$ by $6$ .

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

Step 5.3.3.1.2

Combine $- 6$ and $\frac{6 x - 1}{6 - 5 x}$ .

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

Step 5.3.3.2

Move the negative in front of the fraction.

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

Step 5.3.3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{6 - 5 x}{6 - 5 x}$ .

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

Step 5.3.3.4

Combine $- 1$ and $\frac{6 - 5 x}{6 - 5 x}$ .

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

Step 5.3.3.5

Combine the numerators over the common denominator.

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

Step 5.3.3.6

Rewrite $\frac{- 6 (6 x - 1) - (6 - 5 x)}{6 - 5 x}$ in a factored form.

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Step 5.3.3.6.1

Factor $- 1$ out of $- 6 (6 x - 1) - (6 - 5 x)$ .

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Step 5.3.3.6.1.1

Reorder $6$ and $- 5 x$ .

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

Step 5.3.3.6.1.2

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

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

Step 5.3.3.6.1.3

Factor $- 1$ out of $- (6 (6 x - 1)) - (- 5 x + 6)$ .

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

Step 5.3.3.6.2

Apply the distributive property.

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

Step 5.3.3.6.3

Multiply $6$ by $6$ .

$f (- \frac{6 x - 1}{6 - 5 x}) = \frac{\frac{- (36 x + 6 \cdot - 1 - 5 x + 6)}{6 - 5 x}}{5 (- \frac{6 x - 1}{6 - 5 x}) - 6}$

Step 5.3.3.6.4

Multiply $6$ by $- 1$ .

$f (- \frac{6 x - 1}{6 - 5 x}) = \frac{\frac{- (36 x - 6 - 5 x + 6)}{6 - 5 x}}{5 (- \frac{6 x - 1}{6 - 5 x}) - 6}$

Step 5.3.3.6.5

Subtract $5 x$ from $36 x$ .

$f (- \frac{6 x - 1}{6 - 5 x}) = \frac{\frac{- (31 x - 6 + 6)}{6 - 5 x}}{5 (- \frac{6 x - 1}{6 - 5 x}) - 6}$

Step 5.3.3.6.6

Add $- 6$ and $6$ .

$f (- \frac{6 x - 1}{6 - 5 x}) = \frac{\frac{- (31 x + 0)}{6 - 5 x}}{5 (- \frac{6 x - 1}{6 - 5 x}) - 6}$

Step 5.3.3.6.7

Add $31 x$ and $0$ .

$f (- \frac{6 x - 1}{6 - 5 x}) = \frac{\frac{- 1 \cdot (31 x)}{6 - 5 x}}{5 (- \frac{6 x - 1}{6 - 5 x}) - 6}$

Step 5.3.3.6.8

Multiply $- 1$ by $31$ .

$f (- \frac{6 x - 1}{6 - 5 x}) = \frac{\frac{- 31 x}{6 - 5 x}}{5 (- \frac{6 x - 1}{6 - 5 x}) - 6}$

Step 5.3.3.7

Move the negative in front of the fraction.

$f (- \frac{6 x - 1}{6 - 5 x}) = \frac{- \frac{31 x}{6 - 5 x}}{5 (- \frac{6 x - 1}{6 - 5 x}) - 6}$

Step 5.3.4

Simplify the denominator.

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Step 5.3.4.1

Multiply $5 (- \frac{6 x - 1}{6 - 5 x})$ .

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Step 5.3.4.1.1

Multiply $- 1$ by $5$ .

$f (- \frac{6 x - 1}{6 - 5 x}) = \frac{- \frac{31 x}{6 - 5 x}}{- 5 \frac{6 x - 1}{6 - 5 x} - 6}$

Step 5.3.4.1.2

Combine $- 5$ and $\frac{6 x - 1}{6 - 5 x}$ .

$f (- \frac{6 x - 1}{6 - 5 x}) = \frac{- \frac{31 x}{6 - 5 x}}{\frac{- 5 (6 x - 1)}{6 - 5 x} - 6}$

Step 5.3.4.2

Move the negative in front of the fraction.

$f (- \frac{6 x - 1}{6 - 5 x}) = \frac{- \frac{31 x}{6 - 5 x}}{- \frac{(5) (6 x - 1)}{6 - 5 x} - 6}$

Step 5.3.4.3

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

$f (- \frac{6 x - 1}{6 - 5 x}) = \frac{- \frac{31 x}{6 - 5 x}}{- \frac{5 (6 x - 1)}{6 - 5 x} - 6 \cdot \frac{6 - 5 x}{6 - 5 x}}$

Step 5.3.4.4

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

$f (- \frac{6 x - 1}{6 - 5 x}) = \frac{- \frac{31 x}{6 - 5 x}}{- \frac{5 (6 x - 1)}{6 - 5 x} + \frac{- 6 (6 - 5 x)}{6 - 5 x}}$

Step 5.3.4.5

Combine the numerators over the common denominator.

$f (- \frac{6 x - 1}{6 - 5 x}) = \frac{- \frac{31 x}{6 - 5 x}}{\frac{- 5 (6 x - 1) - 6 (6 - 5 x)}{6 - 5 x}}$

Step 5.3.4.6

Reorder terms.

$f (- \frac{6 x - 1}{6 - 5 x}) = \frac{- \frac{31 x}{6 - 5 x}}{\frac{- 5 (6 x - 1) - 6 (6 - 5 x)}{- 5 x + 6}}$

Step 5.3.4.7

Rewrite $\frac{- 5 (6 x - 1) - 6 (6 - 5 x)}{- 5 x + 6}$ in a factored form.

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Step 5.3.4.7.1

Factor $- 1$ out of $- 5 (6 x - 1) - 6 (6 - 5 x)$ .

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Step 5.3.4.7.1.1

Reorder $6$ and $- 5 x$ .

$f (- \frac{6 x - 1}{6 - 5 x}) = \frac{- \frac{31 x}{6 - 5 x}}{\frac{- 5 (6 x - 1) - 6 (- 5 x + 6)}{- 5 x + 6}}$

Step 5.3.4.7.1.2

Factor $- 1$ out of $- 5 (6 x - 1)$ .

$f (- \frac{6 x - 1}{6 - 5 x}) = \frac{- \frac{31 x}{6 - 5 x}}{\frac{- (5 (6 x - 1)) - 6 (- 5 x + 6)}{- 5 x + 6}}$

Step 5.3.4.7.1.3

Factor $- 1$ out of $- 6 (- 5 x + 6)$ .

$f (- \frac{6 x - 1}{6 - 5 x}) = \frac{- \frac{31 x}{6 - 5 x}}{\frac{- (5 (6 x - 1)) - (6 (- 5 x + 6))}{- 5 x + 6}}$

Step 5.3.4.7.1.4

Factor $- 1$ out of $- (5 (6 x - 1)) - (6 (- 5 x + 6))$ .

$f (- \frac{6 x - 1}{6 - 5 x}) = \frac{- \frac{31 x}{6 - 5 x}}{\frac{- (5 (6 x - 1) + 6 (- 5 x + 6))}{- 5 x + 6}}$

Step 5.3.4.7.2

Apply the distributive property.

$f (- \frac{6 x - 1}{6 - 5 x}) = \frac{- \frac{31 x}{6 - 5 x}}{\frac{- (5 (6 x) + 5 \cdot - 1 + 6 (- 5 x + 6))}{- 5 x + 6}}$

Step 5.3.4.7.3

Multiply $6$ by $5$ .

$f (- \frac{6 x - 1}{6 - 5 x}) = \frac{- \frac{31 x}{6 - 5 x}}{\frac{- (30 x + 5 \cdot - 1 + 6 (- 5 x + 6))}{- 5 x + 6}}$

Step 5.3.4.7.4

Multiply $5$ by $- 1$ .

$f (- \frac{6 x - 1}{6 - 5 x}) = \frac{- \frac{31 x}{6 - 5 x}}{\frac{- (30 x - 5 + 6 (- 5 x + 6))}{- 5 x + 6}}$

Step 5.3.4.7.5

Apply the distributive property.

$f (- \frac{6 x - 1}{6 - 5 x}) = \frac{- \frac{31 x}{6 - 5 x}}{\frac{- (30 x - 5 + 6 (- 5 x) + 6 \cdot 6)}{- 5 x + 6}}$

Step 5.3.4.7.6

Multiply $- 5$ by $6$ .

$f (- \frac{6 x - 1}{6 - 5 x}) = \frac{- \frac{31 x}{6 - 5 x}}{\frac{- (30 x - 5 - 30 x + 6 \cdot 6)}{- 5 x + 6}}$

Step 5.3.4.7.7

Multiply $6$ by $6$ .

$f (- \frac{6 x - 1}{6 - 5 x}) = \frac{- \frac{31 x}{6 - 5 x}}{\frac{- (30 x - 5 - 30 x + 36)}{- 5 x + 6}}$

Step 5.3.4.7.8

Subtract $30 x$ from $30 x$ .

$f (- \frac{6 x - 1}{6 - 5 x}) = \frac{- \frac{31 x}{6 - 5 x}}{\frac{- (0 - 5 + 36)}{- 5 x + 6}}$

Step 5.3.4.7.9

Subtract $5$ from $0$ .

$f (- \frac{6 x - 1}{6 - 5 x}) = \frac{- \frac{31 x}{6 - 5 x}}{\frac{- (- 5 + 36)}{- 5 x + 6}}$

Step 5.3.4.7.10

Add $- 5$ and $36$ .

$f (- \frac{6 x - 1}{6 - 5 x}) = \frac{- \frac{31 x}{6 - 5 x}}{\frac{- 1 \cdot 31}{- 5 x + 6}}$

Step 5.3.4.8

Multiply $- 1$ by $31$ .

$f (- \frac{6 x - 1}{6 - 5 x}) = \frac{- \frac{31 x}{6 - 5 x}}{\frac{- 31}{- 5 x + 6}}$

Step 5.3.4.9

Move the negative in front of the fraction.

$f (- \frac{6 x - 1}{6 - 5 x}) = \frac{- \frac{31 x}{6 - 5 x}}{- \frac{31}{- 5 x + 6}}$

Step 5.3.5

Dividing two negative values results in a positive value.

$f (- \frac{6 x - 1}{6 - 5 x}) = \frac{\frac{31 x}{6 - 5 x}}{\frac{31}{- 5 x + 6}}$

Step 5.3.6

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{6 x - 1}{6 - 5 x}) = \frac{31 x}{6 - 5 x} \cdot \frac{- 5 x + 6}{31}$

Step 5.3.7

Cancel the common factor of $31$ .

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Step 5.3.7.1

Factor $31$ out of $31 x$ .

$f (- \frac{6 x - 1}{6 - 5 x}) = \frac{31 (x)}{6 - 5 x} \cdot \frac{- 5 x + 6}{31}$

Step 5.3.7.2

Cancel the common factor.

$f (- \frac{6 x - 1}{6 - 5 x}) = \frac{31 x}{6 - 5 x} \cdot \frac{- 5 x + 6}{31}$

Step 5.3.7.3

Rewrite the expression.

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

Step 5.3.8

Multiply $\frac{x}{6 - 5 x}$ by $- 5 x + 6$ .

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

Step 5.3.9

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

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Step 5.3.9.1

Reorder terms.

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

Step 5.3.9.2

Cancel the common factor.

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

Step 5.3.9.3

Divide $x$ by $1$ .

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

Step 5.4

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

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