Precalculus Examples

$f (x) = \frac{6 x - 2}{4 x - 5}$

Step 1

Write $f (x) = \frac{6 x - 2}{4 x - 5}$ as an equation.

$y = \frac{6 x - 2}{4 x - 5}$

Step 2

Interchange the variables.

$x = \frac{6 y - 2}{4 y - 5}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{6 y - 2}{4 y - 5} = x$ .

$\frac{6 y - 2}{4 y - 5} = x$

Step 3.2

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

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Step 3.2.1

Factor $2$ out of $6 y$ .

$\frac{2 (3 y) - 2}{4 y - 5} = x$

Step 3.2.2

Factor $2$ out of $- 2$ .

$\frac{2 (3 y) + 2 (- 1)}{4 y - 5} = x$

Step 3.2.3

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

$\frac{2 (3 y - 1)}{4 y - 5} = x$

Step 3.3

Find the LCD of the terms in the equation.

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Step 3.3.1

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

$4 y - 5, 1$

Step 3.3.2

Remove parentheses.

$4 y - 5, 1$

Step 3.3.3

The LCM of one and any expression is the expression.

$4 y - 5$

Step 3.4

Multiply each term in $\frac{2 (3 y - 1)}{4 y - 5} = x$ by $4 y - 5$ to eliminate the fractions.

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Step 3.4.1

Multiply each term in $\frac{2 (3 y - 1)}{4 y - 5} = x$ by $4 y - 5$ .

$\frac{2 (3 y - 1)}{4 y - 5} (4 y - 5) = x (4 y - 5)$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $4 y - 5$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{2 (3 y - 1)}{4 y - 5} (4 y - 5) = x (4 y - 5)$

Step 3.4.2.1.2

Rewrite the expression.

$2 (3 y - 1) = x (4 y - 5)$

Step 3.4.2.2

Apply the distributive property.

$2 (3 y) + 2 \cdot - 1 = x (4 y - 5)$

Step 3.4.2.3

Multiply.

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Step 3.4.2.3.1

Multiply $3$ by $2$ .

$6 y + 2 \cdot - 1 = x (4 y - 5)$

Step 3.4.2.3.2

Multiply $2$ by $- 1$ .

$6 y - 2 = x (4 y - 5)$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Apply the distributive property.

$6 y - 2 = x (4 y) + x \cdot - 5$

Step 3.4.3.2

Reorder.

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Step 3.4.3.2.1

Rewrite using the commutative property of multiplication.

$6 y - 2 = 4 x y + x \cdot - 5$

Step 3.4.3.2.2

Move $- 5$ to the left of $x$ .

$6 y - 2 = 4 x y - 5 x$

Step 3.5

Solve the equation.

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Step 3.5.1

Subtract $4 x y$ from both sides of the equation.

$6 y - 2 - 4 x y = - 5 x$

Step 3.5.2

Add $2$ to both sides of the equation.

$6 y - 4 x y = - 5 x + 2$

Step 3.5.3

Factor $2 y$ out of $6 y - 4 x y$ .

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Step 3.5.3.1

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

$2 y (3) - 4 x y = - 5 x + 2$

Step 3.5.3.2

Factor $2 y$ out of $- 4 x y$ .

$2 y (3) + 2 y (- 2 x) = - 5 x + 2$

Step 3.5.3.3

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

$2 y (3 - 2 x) = - 5 x + 2$

Step 3.5.4

Divide each term in $2 y (3 - 2 x) = - 5 x + 2$ by $2 (3 - 2 x)$ and simplify.

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Step 3.5.4.1

Divide each term in $2 y (3 - 2 x) = - 5 x + 2$ by $2 (3 - 2 x)$ .

$\frac{2 y (3 - 2 x)}{2 (3 - 2 x)} = \frac{- 5 x}{2 (3 - 2 x)} + \frac{2}{2 (3 - 2 x)}$

Step 3.5.4.2

Simplify the left side.

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Step 3.5.4.2.1

Cancel the common factor of $2$ .

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Step 3.5.4.2.1.1

Cancel the common factor.

$\frac{2 y (3 - 2 x)}{2 (3 - 2 x)} = \frac{- 5 x}{2 (3 - 2 x)} + \frac{2}{2 (3 - 2 x)}$

Step 3.5.4.2.1.2

Rewrite the expression.

$\frac{y (3 - 2 x)}{3 - 2 x} = \frac{- 5 x}{2 (3 - 2 x)} + \frac{2}{2 (3 - 2 x)}$

Step 3.5.4.2.2

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

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Step 3.5.4.2.2.1

Cancel the common factor.

$\frac{y (3 - 2 x)}{3 - 2 x} = \frac{- 5 x}{2 (3 - 2 x)} + \frac{2}{2 (3 - 2 x)}$

Step 3.5.4.2.2.2

Divide $y$ by $1$ .

$y = \frac{- 5 x}{2 (3 - 2 x)} + \frac{2}{2 (3 - 2 x)}$

Step 3.5.4.3

Simplify the right side.

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Step 3.5.4.3.1

Simplify each term.

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Step 3.5.4.3.1.1

Move the negative in front of the fraction.

$y = - \frac{(5) x}{2 (3 - 2 x)} + \frac{2}{2 (3 - 2 x)}$

Step 3.5.4.3.1.2

Cancel the common factor of $2$ .

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Step 3.5.4.3.1.2.1

Cancel the common factor.

$y = - \frac{5 x}{2 (3 - 2 x)} + \frac{2}{2 (3 - 2 x)}$

Step 3.5.4.3.1.2.2

Rewrite the expression.

$y = - \frac{5 x}{2 (3 - 2 x)} + \frac{1}{3 - 2 x}$

Step 3.5.4.3.2

To write $\frac{1}{3 - 2 x}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - \frac{5 x}{2 (3 - 2 x)} + \frac{1}{3 - 2 x} \cdot \frac{2}{2}$

Step 3.5.4.3.3

Write each expression with a common denominator of $2 (3 - 2 x)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.5.4.3.3.1

Multiply $\frac{1}{3 - 2 x}$ by $\frac{2}{2}$ .

$y = - \frac{5 x}{2 (3 - 2 x)} + \frac{2}{(3 - 2 x) \cdot 2}$

Step 3.5.4.3.3.2

Reorder the factors of $(3 - 2 x) \cdot 2$ .

$y = - \frac{5 x}{2 (3 - 2 x)} + \frac{2}{2 (3 - 2 x)}$

Step 3.5.4.3.4

Combine the numerators over the common denominator.

$y = \frac{- 5 x + 2}{2 (3 - 2 x)}$

Step 3.5.4.3.5

Factor $- 1$ out of $- 5 x$ .

$y = \frac{- (5 x) + 2}{2 (3 - 2 x)}$

Step 3.5.4.3.6

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

$y = \frac{- (5 x) - 1 (- 2)}{2 (3 - 2 x)}$

Step 3.5.4.3.7

Factor $- 1$ out of $- (5 x) - 1 (- 2)$ .

$y = \frac{- (5 x - 2)}{2 (3 - 2 x)}$

Step 3.5.4.3.8

Simplify the expression.

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Step 3.5.4.3.8.1

Rewrite $- (5 x - 2)$ as $- 1 (5 x - 2)$ .

$y = \frac{- 1 (5 x - 2)}{2 (3 - 2 x)}$

Step 3.5.4.3.8.2

Move the negative in front of the fraction.

$y = - \frac{5 x - 2}{2 (3 - 2 x)}$

Step 4

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

$f^{- 1} (x) = - \frac{5 x - 2}{2 (3 - 2 x)}$

Step 5

Verify if $f^{- 1} (x) = - \frac{5 x - 2}{2 (3 - 2 x)}$ is the inverse of $f (x) = \frac{6 x - 2}{4 x - 5}$ .

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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 $f^{- 1} (\frac{6 x - 2}{4 x - 5})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{5 (\frac{6 x - 2}{4 x - 5}) - 2}{2 (3 - 2 \frac{6 x - 2}{4 x - 5})}$

Step 5.2.3

Simplify with factoring out.

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Step 5.2.3.1

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

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Step 5.2.3.1.1

Factor $2$ out of $6 x$ .

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{5 (\frac{2 (3 x) - 2}{4 x - 5}) - 2}{2 (3 - 2 \frac{6 x - 2}{4 x - 5})}$

Step 5.2.3.1.2

Factor $2$ out of $- 2$ .

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{5 (\frac{2 (3 x) + 2 (- 1)}{4 x - 5}) - 2}{2 (3 - 2 \frac{6 x - 2}{4 x - 5})}$

Step 5.2.3.1.3

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

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{5 (\frac{2 (3 x - 1)}{4 x - 5}) - 2}{2 (3 - 2 \frac{6 x - 2}{4 x - 5})}$

Step 5.2.3.2

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

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Step 5.2.3.2.1

Factor $2$ out of $6 x$ .

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{5 (\frac{2 (3 x - 1)}{4 x - 5}) - 2}{2 (3 - 2 \frac{2 (3 x) - 2}{4 x - 5})}$

Step 5.2.3.2.2

Factor $2$ out of $- 2$ .

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{5 (\frac{2 (3 x - 1)}{4 x - 5}) - 2}{2 (3 - 2 \frac{2 (3 x) + 2 (- 1)}{4 x - 5})}$

Step 5.2.3.2.3

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

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{5 (\frac{2 (3 x - 1)}{4 x - 5}) - 2}{2 (3 - 2 \frac{2 (3 x - 1)}{4 x - 5})}$

Step 5.2.4

Simplify the numerator.

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Step 5.2.4.1

Multiply $5 \frac{2 (3 x - 1)}{4 x - 5}$ .

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Step 5.2.4.1.1

Combine $5$ and $\frac{2 (3 x - 1)}{4 x - 5}$ .

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{\frac{5 (2 (3 x - 1))}{4 x - 5} - 2}{2 (3 - 2 \frac{2 (3 x - 1)}{4 x - 5})}$

Step 5.2.4.1.2

Multiply $2$ by $5$ .

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{\frac{10 (3 x - 1)}{4 x - 5} - 2}{2 (3 - 2 \frac{2 (3 x - 1)}{4 x - 5})}$

Step 5.2.4.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{4 x - 5}{4 x - 5}$ .

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{\frac{10 (3 x - 1)}{4 x - 5} - 2 \cdot \frac{4 x - 5}{4 x - 5}}{2 (3 - 2 \frac{2 (3 x - 1)}{4 x - 5})}$

Step 5.2.4.3

Combine $- 2$ and $\frac{4 x - 5}{4 x - 5}$ .

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{\frac{10 (3 x - 1)}{4 x - 5} + \frac{- 2 (4 x - 5)}{4 x - 5}}{2 (3 - 2 \frac{2 (3 x - 1)}{4 x - 5})}$

Step 5.2.4.4

Combine the numerators over the common denominator.

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{\frac{10 (3 x - 1) - 2 (4 x - 5)}{4 x - 5}}{2 (3 - 2 \frac{2 (3 x - 1)}{4 x - 5})}$

Step 5.2.4.5

Rewrite $\frac{10 (3 x - 1) - 2 (4 x - 5)}{4 x - 5}$ in a factored form.

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Step 5.2.4.5.1

Factor $2$ out of $10 (3 x - 1) - 2 (4 x - 5)$ .

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Step 5.2.4.5.1.1

Factor $2$ out of $10 (3 x - 1)$ .

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{\frac{2 (5 (3 x - 1)) - 2 (4 x - 5)}{4 x - 5}}{2 (3 - 2 \frac{2 (3 x - 1)}{4 x - 5})}$

Step 5.2.4.5.1.2

Factor $2$ out of $- 2 (4 x - 5)$ .

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{\frac{2 (5 (3 x - 1)) + 2 (- (4 x - 5))}{4 x - 5}}{2 (3 - 2 \frac{2 (3 x - 1)}{4 x - 5})}$

Step 5.2.4.5.1.3

Factor $2$ out of $2 (5 (3 x - 1)) + 2 (- (4 x - 5))$ .

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{\frac{2 (5 (3 x - 1) - (4 x - 5))}{4 x - 5}}{2 (3 - 2 \frac{2 (3 x - 1)}{4 x - 5})}$

Step 5.2.4.5.2

Apply the distributive property.

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{\frac{2 (5 (3 x) + 5 \cdot - 1 - (4 x - 5))}{4 x - 5}}{2 (3 - 2 \frac{2 (3 x - 1)}{4 x - 5})}$

Step 5.2.4.5.3

Multiply $3$ by $5$ .

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{\frac{2 (15 x + 5 \cdot - 1 - (4 x - 5))}{4 x - 5}}{2 (3 - 2 \frac{2 (3 x - 1)}{4 x - 5})}$

Step 5.2.4.5.4

Multiply $5$ by $- 1$ .

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{\frac{2 (15 x - 5 - (4 x - 5))}{4 x - 5}}{2 (3 - 2 \frac{2 (3 x - 1)}{4 x - 5})}$

Step 5.2.4.5.5

Apply the distributive property.

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{\frac{2 (15 x - 5 - (4 x) + 5)}{4 x - 5}}{2 (3 - 2 \frac{2 (3 x - 1)}{4 x - 5})}$

Step 5.2.4.5.6

Multiply $4$ by $- 1$ .

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{\frac{2 (15 x - 5 - 4 x + 5)}{4 x - 5}}{2 (3 - 2 \frac{2 (3 x - 1)}{4 x - 5})}$

Step 5.2.4.5.7

Multiply $- 1$ by $- 5$ .

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{\frac{2 (15 x - 5 - 4 x + 5)}{4 x - 5}}{2 (3 - 2 \frac{2 (3 x - 1)}{4 x - 5})}$

Step 5.2.4.5.8

Subtract $4 x$ from $15 x$ .

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{\frac{2 (11 x - 5 + 5)}{4 x - 5}}{2 (3 - 2 \frac{2 (3 x - 1)}{4 x - 5})}$

Step 5.2.4.5.9

Add $- 5$ and $5$ .

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{\frac{2 (11 x + 0)}{4 x - 5}}{2 (3 - 2 \frac{2 (3 x - 1)}{4 x - 5})}$

Step 5.2.4.5.10

Add $11 x$ and $0$ .

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{\frac{2 \cdot (11 x)}{4 x - 5}}{2 (3 - 2 \frac{2 (3 x - 1)}{4 x - 5})}$

Step 5.2.4.5.11

Multiply $2$ by $11$ .

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{\frac{22 x}{4 x - 5}}{2 (3 - 2 \frac{2 (3 x - 1)}{4 x - 5})}$

Step 5.2.5

Simplify the denominator.

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Step 5.2.5.1

Multiply $- 2 \frac{2 (3 x - 1)}{4 x - 5}$ .

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Step 5.2.5.1.1

Combine $- 2$ and $\frac{2 (3 x - 1)}{4 x - 5}$ .

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{\frac{22 x}{4 x - 5}}{2 (3 + \frac{- 2 (2 (3 x - 1))}{4 x - 5})}$

Step 5.2.5.1.2

Multiply $2$ by $- 2$ .

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{\frac{22 x}{4 x - 5}}{2 (3 + \frac{- 4 (3 x - 1)}{4 x - 5})}$

Step 5.2.5.2

Move the negative in front of the fraction.

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{\frac{22 x}{4 x - 5}}{2 (3 - \frac{(4) (3 x - 1)}{4 x - 5})}$

Step 5.2.5.3

To write $3$ as a fraction with a common denominator, multiply by $\frac{4 x - 5}{4 x - 5}$ .

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{\frac{22 x}{4 x - 5}}{2 (\frac{3 (4 x - 5)}{4 x - 5} - \frac{4 (3 x - 1)}{4 x - 5})}$

Step 5.2.5.4

Combine the numerators over the common denominator.

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{\frac{22 x}{4 x - 5}}{2 (\frac{3 (4 x - 5) - 4 (3 x - 1)}{4 x - 5})}$

Step 5.2.5.5

Rewrite $\frac{3 (4 x - 5) - 4 (3 x - 1)}{4 x - 5}$ in a factored form.

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Step 5.2.5.5.1

Apply the distributive property.

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{\frac{22 x}{4 x - 5}}{2 (\frac{3 (4 x) + 3 \cdot - 5 - 4 (3 x - 1)}{4 x - 5})}$

Step 5.2.5.5.2

Multiply $4$ by $3$ .

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{\frac{22 x}{4 x - 5}}{2 (\frac{12 x + 3 \cdot - 5 - 4 (3 x - 1)}{4 x - 5})}$

Step 5.2.5.5.3

Multiply $3$ by $- 5$ .

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{\frac{22 x}{4 x - 5}}{2 (\frac{12 x - 15 - 4 (3 x - 1)}{4 x - 5})}$

Step 5.2.5.5.4

Apply the distributive property.

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{\frac{22 x}{4 x - 5}}{2 (\frac{12 x - 15 - 4 (3 x) - 4 \cdot - 1}{4 x - 5})}$

Step 5.2.5.5.5

Multiply $3$ by $- 4$ .

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{\frac{22 x}{4 x - 5}}{2 (\frac{12 x - 15 - 12 x - 4 \cdot - 1}{4 x - 5})}$

Step 5.2.5.5.6

Multiply $- 4$ by $- 1$ .

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{\frac{22 x}{4 x - 5}}{2 (\frac{12 x - 15 - 12 x + 4}{4 x - 5})}$

Step 5.2.5.5.7

Subtract $12 x$ from $12 x$ .

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{\frac{22 x}{4 x - 5}}{2 (\frac{0 - 15 + 4}{4 x - 5})}$

Step 5.2.5.5.8

Subtract $15$ from $0$ .

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{\frac{22 x}{4 x - 5}}{2 (\frac{- 15 + 4}{4 x - 5})}$

Step 5.2.5.5.9

Add $- 15$ and $4$ .

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{\frac{22 x}{4 x - 5}}{2 (\frac{- 11}{4 x - 5})}$

Step 5.2.5.6

Move the negative in front of the fraction.

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{\frac{22 x}{4 x - 5}}{2 \cdot (- 1 \frac{11}{4 x - 5})}$

Step 5.2.5.7

Combine exponents.

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Step 5.2.5.7.1

Factor out negative.

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{\frac{22 x}{4 x - 5}}{- (2 (\frac{11}{4 x - 5}))}$

Step 5.2.5.7.2

Combine $2$ and $\frac{11}{4 x - 5}$ .

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{\frac{22 x}{4 x - 5}}{- \frac{2 \cdot 11}{4 x - 5}}$

Step 5.2.5.7.3

Multiply $2$ by $11$ .

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - \frac{\frac{22 x}{4 x - 5}}{- \frac{22}{4 x - 5}}$

Step 5.2.6

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - (\frac{22 x}{4 x - 5} \cdot (- \frac{4 x - 5}{22}))$

Step 5.2.7

Rewrite using the commutative property of multiplication.

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - (- \frac{22 x}{4 x - 5} \cdot \frac{4 x - 5}{22})$

Step 5.2.8

Cancel the common factor of $22$ .

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Step 5.2.8.1

Move the leading negative in $- \frac{22 x}{4 x - 5}$ into the numerator.

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - (\frac{- 22 x}{4 x - 5} \cdot \frac{4 x - 5}{22})$

Step 5.2.8.2

Factor $22$ out of $- 22 x$ .

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - (\frac{22 (- x)}{4 x - 5} \cdot \frac{4 x - 5}{22})$

Step 5.2.8.3

Cancel the common factor.

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - (\frac{22 (- x)}{4 x - 5} \cdot \frac{4 x - 5}{22})$

Step 5.2.8.4

Rewrite the expression.

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - (\frac{- x}{4 x - 5} \cdot (4 x - 5))$

Step 5.2.9

Cancel the common factor of $4 x - 5$ .

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Step 5.2.9.1

Cancel the common factor.

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = - (\frac{- x}{4 x - 5} \cdot (4 x - 5))$

Step 5.2.9.2

Rewrite the expression.

$f^{- 1} (\frac{6 x - 2}{4 x - 5}) = x$

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 $f (- \frac{5 x - 2}{2 (3 - 2 x)})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{6 (- \frac{5 x - 2}{2 (3 - 2 x)}) - 2}{4 (- \frac{5 x - 2}{2 (3 - 2 x)}) - 5}$

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Factor $2$ out of $6 (- \frac{5 x - 2}{2 (3 - 2 x)}) - 2$ .

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Step 5.3.3.1.1

Factor $2$ out of $6 (- \frac{5 x - 2}{2 (3 - 2 x)})$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{2 (3 (- \frac{5 x - 2}{2 (3 - 2 x)})) - 2}{4 (- \frac{5 x - 2}{2 (3 - 2 x)}) - 5}$

Step 5.3.3.1.2

Factor $2$ out of $- 2$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{2 (3 (- \frac{5 x - 2}{2 (3 - 2 x)})) + 2 (- 1)}{4 (- \frac{5 x - 2}{2 (3 - 2 x)}) - 5}$

Step 5.3.3.1.3

Factor $2$ out of $2 (3 (- \frac{5 x - 2}{2 (3 - 2 x)})) + 2 (- 1)$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{2 (3 (- \frac{5 x - 2}{2 (3 - 2 x)}) - 1)}{4 (- \frac{5 x - 2}{2 (3 - 2 x)}) - 5}$

Step 5.3.3.2

Multiply $3 (- \frac{5 x - 2}{2 (3 - 2 x)})$ .

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Step 5.3.3.2.1

Multiply $- 1$ by $3$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{2 (- 3 \frac{5 x - 2}{2 (3 - 2 x)} - 1)}{4 (- \frac{5 x - 2}{2 (3 - 2 x)}) - 5}$

Step 5.3.3.2.2

Combine $- 3$ and $\frac{5 x - 2}{2 (3 - 2 x)}$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{2 (\frac{- 3 (5 x - 2)}{2 (3 - 2 x)} - 1)}{4 (- \frac{5 x - 2}{2 (3 - 2 x)}) - 5}$

Step 5.3.3.3

Move the negative in front of the fraction.

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{2 (- \frac{(3) (5 x - 2)}{2 (3 - 2 x)} - 1)}{4 (- \frac{5 x - 2}{2 (3 - 2 x)}) - 5}$

Step 5.3.3.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2 (3 - 2 x)}{2 (3 - 2 x)}$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{2 (- \frac{3 (5 x - 2)}{2 (3 - 2 x)} - 1 \cdot \frac{2 (3 - 2 x)}{2 (3 - 2 x)})}{4 (- \frac{5 x - 2}{2 (3 - 2 x)}) - 5}$

Step 5.3.3.5

Combine $- 1$ and $\frac{2 (3 - 2 x)}{2 (3 - 2 x)}$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{2 (- \frac{3 (5 x - 2)}{2 (3 - 2 x)} + \frac{- (2 (3 - 2 x))}{2 (3 - 2 x)})}{4 (- \frac{5 x - 2}{2 (3 - 2 x)}) - 5}$

Step 5.3.3.6

Combine the numerators over the common denominator.

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{2 (\frac{- 3 (5 x - 2) - (2 (3 - 2 x))}{2 (3 - 2 x)})}{4 (- \frac{5 x - 2}{2 (3 - 2 x)}) - 5}$

Step 5.3.3.7

Reorder terms.

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{2 (\frac{- 1 \cdot (2 (3 - 2 x)) - 3 (5 x - 2)}{2 (3 - 2 x)})}{4 (- \frac{5 x - 2}{2 (3 - 2 x)}) - 5}$

Step 5.3.3.8

Rewrite $\frac{- 1 \cdot 2 (3 - 2 x) - 3 (5 x - 2)}{2 (3 - 2 x)}$ in a factored form.

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Step 5.3.3.8.1

Factor $- 1$ out of $- 1 \cdot 2 (3 - 2 x) - 3 (5 x - 2)$ .

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Step 5.3.3.8.1.1

Reorder $3$ and $- 2 x$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{2 (\frac{- 1 \cdot (2 (- 2 x + 3)) - 3 (5 x - 2)}{2 (3 - 2 x)})}{4 (- \frac{5 x - 2}{2 (3 - 2 x)}) - 5}$

Step 5.3.3.8.1.2

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

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{2 (\frac{- (2 (- 2 x + 3)) - 3 (5 x - 2)}{2 (3 - 2 x)})}{4 (- \frac{5 x - 2}{2 (3 - 2 x)}) - 5}$

Step 5.3.3.8.1.3

Factor $- 1$ out of $- 3 (5 x - 2)$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{2 (\frac{- (2 (- 2 x + 3)) - (3 (5 x - 2))}{2 (3 - 2 x)})}{4 (- \frac{5 x - 2}{2 (3 - 2 x)}) - 5}$

Step 5.3.3.8.1.4

Factor $- 1$ out of $- (2 (- 2 x + 3)) - (3 (5 x - 2))$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{2 (\frac{- (2 (- 2 x + 3) + 3 (5 x - 2))}{2 (3 - 2 x)})}{4 (- \frac{5 x - 2}{2 (3 - 2 x)}) - 5}$

Step 5.3.3.8.2

Apply the distributive property.

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{2 (\frac{- (2 (- 2 x) + 2 \cdot 3 + 3 (5 x - 2))}{2 (3 - 2 x)})}{4 (- \frac{5 x - 2}{2 (3 - 2 x)}) - 5}$

Step 5.3.3.8.3

Multiply $- 2$ by $2$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{2 (\frac{- (- 4 x + 2 \cdot 3 + 3 (5 x - 2))}{2 (3 - 2 x)})}{4 (- \frac{5 x - 2}{2 (3 - 2 x)}) - 5}$

Step 5.3.3.8.4

Multiply $2$ by $3$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{2 (\frac{- (- 4 x + 6 + 3 (5 x - 2))}{2 (3 - 2 x)})}{4 (- \frac{5 x - 2}{2 (3 - 2 x)}) - 5}$

Step 5.3.3.8.5

Apply the distributive property.

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{2 (\frac{- (- 4 x + 6 + 3 (5 x) + 3 \cdot - 2)}{2 (3 - 2 x)})}{4 (- \frac{5 x - 2}{2 (3 - 2 x)}) - 5}$

Step 5.3.3.8.6

Multiply $5$ by $3$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{2 (\frac{- (- 4 x + 6 + 15 x + 3 \cdot - 2)}{2 (3 - 2 x)})}{4 (- \frac{5 x - 2}{2 (3 - 2 x)}) - 5}$

Step 5.3.3.8.7

Multiply $3$ by $- 2$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{2 (\frac{- (- 4 x + 6 + 15 x - 6)}{2 (3 - 2 x)})}{4 (- \frac{5 x - 2}{2 (3 - 2 x)}) - 5}$

Step 5.3.3.8.8

Add $- 4 x$ and $15 x$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{2 (\frac{- (11 x + 6 - 6)}{2 (3 - 2 x)})}{4 (- \frac{5 x - 2}{2 (3 - 2 x)}) - 5}$

Step 5.3.3.8.9

Subtract $6$ from $6$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{2 (\frac{- (11 x + 0)}{2 (3 - 2 x)})}{4 (- \frac{5 x - 2}{2 (3 - 2 x)}) - 5}$

Step 5.3.3.8.10

Add $11 x$ and $0$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{2 (\frac{- 1 \cdot (11 x)}{2 (3 - 2 x)})}{4 (- \frac{5 x - 2}{2 (3 - 2 x)}) - 5}$

Step 5.3.3.8.11

Multiply $- 1$ by $11$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{2 (\frac{- 11 x}{2 (3 - 2 x)})}{4 (- \frac{5 x - 2}{2 (3 - 2 x)}) - 5}$

Step 5.3.3.9

Move the negative in front of the fraction.

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{2 (- \frac{(11) x}{2 (3 - 2 x)})}{4 (- \frac{5 x - 2}{2 (3 - 2 x)}) - 5}$

Step 5.3.3.10

Remove unnecessary parentheses.

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{2 \cdot (- 1 \frac{11 x}{2 (3 - 2 x)})}{4 (- \frac{5 x - 2}{2 (3 - 2 x)}) - 5}$

Step 5.3.3.11

Combine exponents.

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Step 5.3.3.11.1

Factor out negative.

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{- (2 (\frac{11 x}{2 (3 - 2 x)}))}{4 (- \frac{5 x - 2}{2 (3 - 2 x)}) - 5}$

Step 5.3.3.11.2

Combine $2$ and $\frac{11 x}{2 (3 - 2 x)}$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{- \frac{2 (11 x)}{2 (3 - 2 x)}}{4 (- \frac{5 x - 2}{2 (3 - 2 x)}) - 5}$

Step 5.3.3.11.3

Multiply $11$ by $2$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{- \frac{22 x}{2 (3 - 2 x)}}{4 (- \frac{5 x - 2}{2 (3 - 2 x)}) - 5}$

Step 5.3.3.12

Reduce the expression $\frac{22 x}{2 (3 - 2 x)}$ by cancelling the common factors.

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Step 5.3.3.12.1

Factor $2$ out of $22 x$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{- \frac{2 (11 x)}{2 (3 - 2 x)}}{4 (- \frac{5 x - 2}{2 (3 - 2 x)}) - 5}$

Step 5.3.3.12.2

Cancel the common factor.

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{- \frac{2 (11 x)}{2 (3 - 2 x)}}{4 (- \frac{5 x - 2}{2 (3 - 2 x)}) - 5}$

Step 5.3.3.12.3

Rewrite the expression.

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{- \frac{11 x}{3 - 2 x}}{4 (- \frac{5 x - 2}{2 (3 - 2 x)}) - 5}$

Step 5.3.4

Simplify the denominator.

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Step 5.3.4.1

Cancel the common factor of $2$ .

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Step 5.3.4.1.1

Move the leading negative in $- \frac{5 x - 2}{2 (3 - 2 x)}$ into the numerator.

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{- \frac{11 x}{3 - 2 x}}{4 (\frac{- (5 x - 2)}{2 (3 - 2 x)}) - 5}$

Step 5.3.4.1.2

Factor $2$ out of $4$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{- \frac{11 x}{3 - 2 x}}{2 (2) (\frac{- (5 x - 2)}{2 (3 - 2 x)}) - 5}$

Step 5.3.4.1.3

Cancel the common factor.

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{- \frac{11 x}{3 - 2 x}}{2 \cdot (2 (\frac{- (5 x - 2)}{2 (3 - 2 x)})) - 5}$

Step 5.3.4.1.4

Rewrite the expression.

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{- \frac{11 x}{3 - 2 x}}{2 (\frac{- (5 x - 2)}{3 - 2 x}) - 5}$

Step 5.3.4.2

Combine $2$ and $\frac{- (5 x - 2)}{3 - 2 x}$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{- \frac{11 x}{3 - 2 x}}{\frac{2 (- (5 x - 2))}{3 - 2 x} - 5}$

Step 5.3.4.3

Multiply $- 1$ by $2$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{- \frac{11 x}{3 - 2 x}}{\frac{- 2 (5 x - 2)}{3 - 2 x} - 5}$

Step 5.3.4.4

Move the negative in front of the fraction.

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{- \frac{11 x}{3 - 2 x}}{- \frac{(2) (5 x - 2)}{3 - 2 x} - 5}$

Step 5.3.4.5

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{3 - 2 x}{3 - 2 x}$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{- \frac{11 x}{3 - 2 x}}{- \frac{2 (5 x - 2)}{3 - 2 x} - 5 \cdot \frac{3 - 2 x}{3 - 2 x}}$

Step 5.3.4.6

Combine $- 5$ and $\frac{3 - 2 x}{3 - 2 x}$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{- \frac{11 x}{3 - 2 x}}{- \frac{2 (5 x - 2)}{3 - 2 x} + \frac{- 5 (3 - 2 x)}{3 - 2 x}}$

Step 5.3.4.7

Combine the numerators over the common denominator.

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{- \frac{11 x}{3 - 2 x}}{\frac{- 2 (5 x - 2) - 5 (3 - 2 x)}{3 - 2 x}}$

Step 5.3.4.8

Reorder terms.

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{- \frac{11 x}{3 - 2 x}}{\frac{- 2 (5 x - 2) - 5 (3 - 2 x)}{- 2 x + 3}}$

Step 5.3.4.9

Rewrite $\frac{- 2 (5 x - 2) - 5 (3 - 2 x)}{- 2 x + 3}$ in a factored form.

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Step 5.3.4.9.1

Factor $- 1$ out of $- 2 (5 x - 2) - 5 (3 - 2 x)$ .

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Step 5.3.4.9.1.1

Reorder $3$ and $- 2 x$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{- \frac{11 x}{3 - 2 x}}{\frac{- 2 (5 x - 2) - 5 (- 2 x + 3)}{- 2 x + 3}}$

Step 5.3.4.9.1.2

Factor $- 1$ out of $- 2 (5 x - 2)$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{- \frac{11 x}{3 - 2 x}}{\frac{- (2 (5 x - 2)) - 5 (- 2 x + 3)}{- 2 x + 3}}$

Step 5.3.4.9.1.3

Factor $- 1$ out of $- 5 (- 2 x + 3)$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{- \frac{11 x}{3 - 2 x}}{\frac{- (2 (5 x - 2)) - (5 (- 2 x + 3))}{- 2 x + 3}}$

Step 5.3.4.9.1.4

Factor $- 1$ out of $- (2 (5 x - 2)) - (5 (- 2 x + 3))$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{- \frac{11 x}{3 - 2 x}}{\frac{- (2 (5 x - 2) + 5 (- 2 x + 3))}{- 2 x + 3}}$

Step 5.3.4.9.2

Apply the distributive property.

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{- \frac{11 x}{3 - 2 x}}{\frac{- (2 (5 x) + 2 \cdot - 2 + 5 (- 2 x + 3))}{- 2 x + 3}}$

Step 5.3.4.9.3

Multiply $5$ by $2$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{- \frac{11 x}{3 - 2 x}}{\frac{- (10 x + 2 \cdot - 2 + 5 (- 2 x + 3))}{- 2 x + 3}}$

Step 5.3.4.9.4

Multiply $2$ by $- 2$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{- \frac{11 x}{3 - 2 x}}{\frac{- (10 x - 4 + 5 (- 2 x + 3))}{- 2 x + 3}}$

Step 5.3.4.9.5

Apply the distributive property.

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{- \frac{11 x}{3 - 2 x}}{\frac{- (10 x - 4 + 5 (- 2 x) + 5 \cdot 3)}{- 2 x + 3}}$

Step 5.3.4.9.6

Multiply $- 2$ by $5$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{- \frac{11 x}{3 - 2 x}}{\frac{- (10 x - 4 - 10 x + 5 \cdot 3)}{- 2 x + 3}}$

Step 5.3.4.9.7

Multiply $5$ by $3$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{- \frac{11 x}{3 - 2 x}}{\frac{- (10 x - 4 - 10 x + 15)}{- 2 x + 3}}$

Step 5.3.4.9.8

Subtract $10 x$ from $10 x$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{- \frac{11 x}{3 - 2 x}}{\frac{- (0 - 4 + 15)}{- 2 x + 3}}$

Step 5.3.4.9.9

Subtract $4$ from $0$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{- \frac{11 x}{3 - 2 x}}{\frac{- (- 4 + 15)}{- 2 x + 3}}$

Step 5.3.4.9.10

Add $- 4$ and $15$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{- \frac{11 x}{3 - 2 x}}{\frac{- 1 \cdot 11}{- 2 x + 3}}$

Step 5.3.4.10

Multiply $- 1$ by $11$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{- \frac{11 x}{3 - 2 x}}{\frac{- 11}{- 2 x + 3}}$

Step 5.3.4.11

Move the negative in front of the fraction.

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{- \frac{11 x}{3 - 2 x}}{- \frac{11}{- 2 x + 3}}$

Step 5.3.5

Dividing two negative values results in a positive value.

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{\frac{11 x}{3 - 2 x}}{\frac{11}{- 2 x + 3}}$

Step 5.3.6

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{11 x}{3 - 2 x} \cdot \frac{- 2 x + 3}{11}$

Step 5.3.7

Cancel the common factor of $11$ .

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Step 5.3.7.1

Factor $11$ out of $11 x$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{11 (x)}{3 - 2 x} \cdot \frac{- 2 x + 3}{11}$

Step 5.3.7.2

Cancel the common factor.

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{11 x}{3 - 2 x} \cdot \frac{- 2 x + 3}{11}$

Step 5.3.7.3

Rewrite the expression.

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{x}{3 - 2 x} \cdot (- 2 x + 3)$

Step 5.3.8

Multiply $\frac{x}{3 - 2 x}$ by $- 2 x + 3$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{x (- 2 x + 3)}{3 - 2 x}$

Step 5.3.9

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

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Step 5.3.9.1

Reorder terms.

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{x (- 2 x + 3)}{- 2 x + 3}$

Step 5.3.9.2

Cancel the common factor.

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = \frac{x (- 2 x + 3)}{- 2 x + 3}$

Step 5.3.9.3

Divide $x$ by $1$ .

$f (- \frac{5 x - 2}{2 (3 - 2 x)}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{5 x - 2}{2 (3 - 2 x)}$ is the inverse of $f (x) = \frac{6 x - 2}{4 x - 5}$ .

$f^{- 1} (x) = - \frac{5 x - 2}{2 (3 - 2 x)}$