Precalculus Examples

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

Step 1

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

$y = \frac{6 x + 4}{4 x + 5}$

Step 2

Interchange the variables.

$x = \frac{6 y + 4}{4 y + 5}$

Step 3

Solve for $y$ .

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Step 3.1

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

$\frac{6 y + 4}{4 y + 5} = x$

Step 3.2

Factor $2$ out of $6 y + 4$ .

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Step 3.2.1

Factor $2$ out of $6 y$ .

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

Step 3.2.2

Factor $2$ out of $4$ .

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

Step 3.2.3

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

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

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 $\frac{2 (3 y + 2)}{4 y + 5} = x$ by $4 y + 5$ to eliminate the fractions.

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Step 3.4.1

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

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

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.

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

Step 3.4.2.1.2

Rewrite the expression.

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

Step 3.4.2.2

Apply the distributive property.

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

Step 3.4.2.3

Multiply.

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Step 3.4.2.3.1

Multiply $3$ by $2$ .

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

Step 3.4.2.3.2

Multiply $2$ by $2$ .

$6 y + 4 = x (4 y + 5)$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Apply the distributive property.

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

Step 3.4.3.2

Reorder.

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Step 3.4.3.2.1

Rewrite using the commutative property of multiplication.

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

Step 3.4.3.2.2

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

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

Step 3.5

Solve the equation.

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Step 3.5.1

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

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

Step 3.5.2

Subtract $4$ from both sides of the equation.

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

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 $6 y$ .

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

Step 3.5.3.2

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

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

Step 3.5.3.3

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

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

Step 3.5.4

Divide each term in $2 y (3 - 2 x) = 5 x - 4$ by $2 (3 - 2 x)$ and simplify.

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Step 3.5.4.1

Divide each term in $2 y (3 - 2 x) = 5 x - 4$ by $2 (3 - 2 x)$ .

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

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.

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

Step 3.5.4.2.1.2

Rewrite the expression.

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

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.

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

Step 3.5.4.2.2.2

Divide $y$ by $1$ .

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

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

Cancel the common factor of $- 4$ and $2$ .

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Step 3.5.4.3.1.1.1

Factor $2$ out of $- 4$ .

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

Step 3.5.4.3.1.1.2

Cancel the common factors.

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Step 3.5.4.3.1.1.2.1

Cancel the common factor.

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

Step 3.5.4.3.1.1.2.2

Rewrite the expression.

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

Step 3.5.4.3.1.2

Move the negative in front of the fraction.

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

Step 3.5.4.3.2

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

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

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

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

Step 3.5.4.3.3.2

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

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

Step 3.5.4.3.4

Combine the numerators over the common denominator.

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

Step 3.5.4.3.5

Multiply $- 2$ by $2$ .

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

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Simplify with factoring out.

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Step 5.2.3.1

Factor $2$ out of $6 x + 4$ .

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Step 5.2.3.1.1

Factor $2$ out of $6 x$ .

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

Step 5.2.3.1.2

Factor $2$ out of $4$ .

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

Step 5.2.3.1.3

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

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

Step 5.2.3.2

Factor $2$ out of $6 x + 4$ .

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Step 5.2.3.2.1

Factor $2$ out of $6 x$ .

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

Step 5.2.3.2.2

Factor $2$ out of $4$ .

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

Step 5.2.3.2.3

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

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

Step 5.2.4

Simplify the numerator.

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Step 5.2.4.1

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

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Step 5.2.4.1.1

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

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

Step 5.2.4.1.2

Multiply $2$ by $5$ .

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

Step 5.2.4.2

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

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

Step 5.2.4.3

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

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

Step 5.2.4.4

Combine the numerators over the common denominator.

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

Step 5.2.4.5

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

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Step 5.2.4.5.1

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

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Step 5.2.4.5.1.1

Factor $2$ out of $10 (3 x + 2)$ .

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

Step 5.2.4.5.1.2

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

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

Step 5.2.4.5.1.3

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

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

Step 5.2.4.5.2

Apply the distributive property.

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

Step 5.2.4.5.3

Multiply $3$ by $5$ .

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

Step 5.2.4.5.4

Multiply $5$ by $2$ .

$f^{- 1} (\frac{6 x + 4}{4 x + 5}) = \frac{\frac{2 (15 x + 10 - 2 (4 x + 5))}{4 x + 5}}{2 (3 - 2 \frac{2 (3 x + 2)}{4 x + 5})}$

Step 5.2.4.5.5

Apply the distributive property.

$f^{- 1} (\frac{6 x + 4}{4 x + 5}) = \frac{\frac{2 (15 x + 10 - 2 (4 x) - 2 \cdot 5)}{4 x + 5}}{2 (3 - 2 \frac{2 (3 x + 2)}{4 x + 5})}$

Step 5.2.4.5.6

Multiply $4$ by $- 2$ .

$f^{- 1} (\frac{6 x + 4}{4 x + 5}) = \frac{\frac{2 (15 x + 10 - 8 x - 2 \cdot 5)}{4 x + 5}}{2 (3 - 2 \frac{2 (3 x + 2)}{4 x + 5})}$

Step 5.2.4.5.7

Multiply $- 2$ by $5$ .

$f^{- 1} (\frac{6 x + 4}{4 x + 5}) = \frac{\frac{2 (15 x + 10 - 8 x - 10)}{4 x + 5}}{2 (3 - 2 \frac{2 (3 x + 2)}{4 x + 5})}$

Step 5.2.4.5.8

Subtract $8 x$ from $15 x$ .

$f^{- 1} (\frac{6 x + 4}{4 x + 5}) = \frac{\frac{2 (7 x + 10 - 10)}{4 x + 5}}{2 (3 - 2 \frac{2 (3 x + 2)}{4 x + 5})}$

Step 5.2.4.5.9

Subtract $10$ from $10$ .

$f^{- 1} (\frac{6 x + 4}{4 x + 5}) = \frac{\frac{2 (7 x + 0)}{4 x + 5}}{2 (3 - 2 \frac{2 (3 x + 2)}{4 x + 5})}$

Step 5.2.4.5.10

Add $7 x$ and $0$ .

$f^{- 1} (\frac{6 x + 4}{4 x + 5}) = \frac{\frac{2 \cdot (7 x)}{4 x + 5}}{2 (3 - 2 \frac{2 (3 x + 2)}{4 x + 5})}$

Step 5.2.4.5.11

Multiply $2$ by $7$ .

$f^{- 1} (\frac{6 x + 4}{4 x + 5}) = \frac{\frac{14 x}{4 x + 5}}{2 (3 - 2 \frac{2 (3 x + 2)}{4 x + 5})}$

Step 5.2.5

Simplify the denominator.

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Step 5.2.5.1

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

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Step 5.2.5.1.1

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

$f^{- 1} (\frac{6 x + 4}{4 x + 5}) = \frac{\frac{14 x}{4 x + 5}}{2 (3 + \frac{- 2 (2 (3 x + 2))}{4 x + 5})}$

Step 5.2.5.1.2

Multiply $2$ by $- 2$ .

$f^{- 1} (\frac{6 x + 4}{4 x + 5}) = \frac{\frac{14 x}{4 x + 5}}{2 (3 + \frac{- 4 (3 x + 2)}{4 x + 5})}$

Step 5.2.5.2

Move the negative in front of the fraction.

$f^{- 1} (\frac{6 x + 4}{4 x + 5}) = \frac{\frac{14 x}{4 x + 5}}{2 (3 - \frac{(4) (3 x + 2)}{4 x + 5})}$

Step 5.2.5.3

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

$f^{- 1} (\frac{6 x + 4}{4 x + 5}) = \frac{\frac{14 x}{4 x + 5}}{2 (\frac{3 (4 x + 5)}{4 x + 5} - \frac{4 (3 x + 2)}{4 x + 5})}$

Step 5.2.5.4

Combine the numerators over the common denominator.

$f^{- 1} (\frac{6 x + 4}{4 x + 5}) = \frac{\frac{14 x}{4 x + 5}}{2 (\frac{3 (4 x + 5) - 4 (3 x + 2)}{4 x + 5})}$

Step 5.2.5.5

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

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Step 5.2.5.5.1

Apply the distributive property.

$f^{- 1} (\frac{6 x + 4}{4 x + 5}) = \frac{\frac{14 x}{4 x + 5}}{2 (\frac{3 (4 x) + 3 \cdot 5 - 4 (3 x + 2)}{4 x + 5})}$

Step 5.2.5.5.2

Multiply $4$ by $3$ .

$f^{- 1} (\frac{6 x + 4}{4 x + 5}) = \frac{\frac{14 x}{4 x + 5}}{2 (\frac{12 x + 3 \cdot 5 - 4 (3 x + 2)}{4 x + 5})}$

Step 5.2.5.5.3

Multiply $3$ by $5$ .

$f^{- 1} (\frac{6 x + 4}{4 x + 5}) = \frac{\frac{14 x}{4 x + 5}}{2 (\frac{12 x + 15 - 4 (3 x + 2)}{4 x + 5})}$

Step 5.2.5.5.4

Apply the distributive property.

$f^{- 1} (\frac{6 x + 4}{4 x + 5}) = \frac{\frac{14 x}{4 x + 5}}{2 (\frac{12 x + 15 - 4 (3 x) - 4 \cdot 2}{4 x + 5})}$

Step 5.2.5.5.5

Multiply $3$ by $- 4$ .

$f^{- 1} (\frac{6 x + 4}{4 x + 5}) = \frac{\frac{14 x}{4 x + 5}}{2 (\frac{12 x + 15 - 12 x - 4 \cdot 2}{4 x + 5})}$

Step 5.2.5.5.6

Multiply $- 4$ by $2$ .

$f^{- 1} (\frac{6 x + 4}{4 x + 5}) = \frac{\frac{14 x}{4 x + 5}}{2 (\frac{12 x + 15 - 12 x - 8}{4 x + 5})}$

Step 5.2.5.5.7

Subtract $12 x$ from $12 x$ .

$f^{- 1} (\frac{6 x + 4}{4 x + 5}) = \frac{\frac{14 x}{4 x + 5}}{2 (\frac{0 + 15 - 8}{4 x + 5})}$

Step 5.2.5.5.8

Add $0$ and $15$ .

$f^{- 1} (\frac{6 x + 4}{4 x + 5}) = \frac{\frac{14 x}{4 x + 5}}{2 (\frac{15 - 8}{4 x + 5})}$

Step 5.2.5.5.9

Subtract $8$ from $15$ .

$f^{- 1} (\frac{6 x + 4}{4 x + 5}) = \frac{\frac{14 x}{4 x + 5}}{2 (\frac{7}{4 x + 5})}$

Step 5.2.6

Combine fractions.

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Step 5.2.6.1

Combine $2$ and $\frac{7}{4 x + 5}$ .

$f^{- 1} (\frac{6 x + 4}{4 x + 5}) = \frac{\frac{14 x}{4 x + 5}}{\frac{2 \cdot 7}{4 x + 5}}$

Step 5.2.6.2

Multiply $2$ by $7$ .

$f^{- 1} (\frac{6 x + 4}{4 x + 5}) = \frac{\frac{14 x}{4 x + 5}}{\frac{14}{4 x + 5}}$

Step 5.2.7

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{6 x + 4}{4 x + 5}) = \frac{14 x}{4 x + 5} \cdot \frac{4 x + 5}{14}$

Step 5.2.8

Cancel the common factor of $14$ .

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Step 5.2.8.1

Factor $14$ out of $14 x$ .

$f^{- 1} (\frac{6 x + 4}{4 x + 5}) = \frac{14 (x)}{4 x + 5} \cdot \frac{4 x + 5}{14}$

Step 5.2.8.2

Cancel the common factor.

$f^{- 1} (\frac{6 x + 4}{4 x + 5}) = \frac{14 x}{4 x + 5} \cdot \frac{4 x + 5}{14}$

Step 5.2.8.3

Rewrite the expression.

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

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.

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

Step 5.2.9.2

Rewrite the expression.

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

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 $f (\frac{5 x - 4}{2 (3 - 2 x)})$ by substituting in the value of $f^{- 1}$ into $f$ .

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

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Factor $2$ out of $6 \frac{5 x - 4}{2 (3 - 2 x)} + 4$ .

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Step 5.3.3.1.1

Factor $2$ out of $6 \frac{5 x - 4}{2 (3 - 2 x)}$ .

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

Step 5.3.3.1.2

Factor $2$ out of $4$ .

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

Step 5.3.3.1.3

Factor $2$ out of $2 (3 \frac{5 x - 4}{2 (3 - 2 x)}) + 2 (2)$ .

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

Step 5.3.3.2

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

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

Step 5.3.3.3

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

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

Step 5.3.3.4

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

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

Step 5.3.3.5

Combine the numerators over the common denominator.

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

Step 5.3.3.6

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

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Step 5.3.3.6.1

Apply the distributive property.

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

Step 5.3.3.6.2

Multiply $5$ by $3$ .

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

Step 5.3.3.6.3

Multiply $3$ by $- 4$ .

$f (\frac{5 x - 4}{2 (3 - 2 x)}) = \frac{2 (\frac{15 x - 12 + 2 (2 (3 - 2 x))}{2 (3 - 2 x)})}{4 (\frac{5 x - 4}{2 (3 - 2 x)}) + 5}$

Step 5.3.3.6.4

Apply the distributive property.

$f (\frac{5 x - 4}{2 (3 - 2 x)}) = \frac{2 (\frac{15 x - 12 + 2 (2 \cdot 3 + 2 (- 2 x))}{2 (3 - 2 x)})}{4 (\frac{5 x - 4}{2 (3 - 2 x)}) + 5}$

Step 5.3.3.6.5

Multiply $2$ by $3$ .

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

Step 5.3.3.6.6

Multiply $- 2$ by $2$ .

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

Step 5.3.3.6.7

Apply the distributive property.

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

Step 5.3.3.6.8

Multiply $2$ by $6$ .

$f (\frac{5 x - 4}{2 (3 - 2 x)}) = \frac{2 (\frac{15 x - 12 + 12 + 2 (- 4 x)}{2 (3 - 2 x)})}{4 (\frac{5 x - 4}{2 (3 - 2 x)}) + 5}$

Step 5.3.3.6.9

Multiply $- 4$ by $2$ .

$f (\frac{5 x - 4}{2 (3 - 2 x)}) = \frac{2 (\frac{15 x - 12 + 12 - 8 x}{2 (3 - 2 x)})}{4 (\frac{5 x - 4}{2 (3 - 2 x)}) + 5}$

Step 5.3.3.6.10

Subtract $8 x$ from $15 x$ .

$f (\frac{5 x - 4}{2 (3 - 2 x)}) = \frac{2 (\frac{7 x - 12 + 12}{2 (3 - 2 x)})}{4 (\frac{5 x - 4}{2 (3 - 2 x)}) + 5}$

Step 5.3.3.6.11

Add $- 12$ and $12$ .

$f (\frac{5 x - 4}{2 (3 - 2 x)}) = \frac{2 (\frac{7 x + 0}{2 (3 - 2 x)})}{4 (\frac{5 x - 4}{2 (3 - 2 x)}) + 5}$

Step 5.3.3.6.12

Add $7 x$ and $0$ .

$f (\frac{5 x - 4}{2 (3 - 2 x)}) = \frac{2 (\frac{7 x}{2 (3 - 2 x)})}{4 (\frac{5 x - 4}{2 (3 - 2 x)}) + 5}$

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

Factor $2$ out of $4$ .

$f (\frac{5 x - 4}{2 (3 - 2 x)}) = \frac{2 (\frac{7 x}{2 (3 - 2 x)})}{2 (2) (\frac{5 x - 4}{2 (3 - 2 x)}) + 5}$

Step 5.3.4.1.2

Cancel the common factor.

$f (\frac{5 x - 4}{2 (3 - 2 x)}) = \frac{2 (\frac{7 x}{2 (3 - 2 x)})}{2 \cdot (2 (\frac{5 x - 4}{2 (3 - 2 x)})) + 5}$

Step 5.3.4.1.3

Rewrite the expression.

$f (\frac{5 x - 4}{2 (3 - 2 x)}) = \frac{2 (\frac{7 x}{2 (3 - 2 x)})}{2 (\frac{5 x - 4}{3 - 2 x}) + 5}$

Step 5.3.4.2

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

$f (\frac{5 x - 4}{2 (3 - 2 x)}) = \frac{2 (\frac{7 x}{2 (3 - 2 x)})}{\frac{2 (5 x - 4)}{3 - 2 x} + 5}$

Step 5.3.4.3

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

$f (\frac{5 x - 4}{2 (3 - 2 x)}) = \frac{2 (\frac{7 x}{2 (3 - 2 x)})}{\frac{2 (5 x - 4)}{3 - 2 x} + \frac{5 (3 - 2 x)}{3 - 2 x}}$

Step 5.3.4.4

Combine the numerators over the common denominator.

$f (\frac{5 x - 4}{2 (3 - 2 x)}) = \frac{2 (\frac{7 x}{2 (3 - 2 x)})}{\frac{2 (5 x - 4) + 5 (3 - 2 x)}{3 - 2 x}}$

Step 5.3.4.5

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

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Step 5.3.4.5.1

Apply the distributive property.

$f (\frac{5 x - 4}{2 (3 - 2 x)}) = \frac{2 (\frac{7 x}{2 (3 - 2 x)})}{\frac{2 (5 x) + 2 \cdot - 4 + 5 (3 - 2 x)}{3 - 2 x}}$

Step 5.3.4.5.2

Multiply $5$ by $2$ .

$f (\frac{5 x - 4}{2 (3 - 2 x)}) = \frac{2 (\frac{7 x}{2 (3 - 2 x)})}{\frac{10 x + 2 \cdot - 4 + 5 (3 - 2 x)}{3 - 2 x}}$

Step 5.3.4.5.3

Multiply $2$ by $- 4$ .

$f (\frac{5 x - 4}{2 (3 - 2 x)}) = \frac{2 (\frac{7 x}{2 (3 - 2 x)})}{\frac{10 x - 8 + 5 (3 - 2 x)}{3 - 2 x}}$

Step 5.3.4.5.4

Apply the distributive property.

$f (\frac{5 x - 4}{2 (3 - 2 x)}) = \frac{2 (\frac{7 x}{2 (3 - 2 x)})}{\frac{10 x - 8 + 5 \cdot 3 + 5 (- 2 x)}{3 - 2 x}}$

Step 5.3.4.5.5

Multiply $5$ by $3$ .

$f (\frac{5 x - 4}{2 (3 - 2 x)}) = \frac{2 (\frac{7 x}{2 (3 - 2 x)})}{\frac{10 x - 8 + 15 + 5 (- 2 x)}{3 - 2 x}}$

Step 5.3.4.5.6

Multiply $- 2$ by $5$ .

$f (\frac{5 x - 4}{2 (3 - 2 x)}) = \frac{2 (\frac{7 x}{2 (3 - 2 x)})}{\frac{10 x - 8 + 15 - 10 x}{3 - 2 x}}$

Step 5.3.4.5.7

Subtract $10 x$ from $10 x$ .

$f (\frac{5 x - 4}{2 (3 - 2 x)}) = \frac{2 (\frac{7 x}{2 (3 - 2 x)})}{\frac{0 - 8 + 15}{3 - 2 x}}$

Step 5.3.4.5.8

Subtract $8$ from $0$ .

$f (\frac{5 x - 4}{2 (3 - 2 x)}) = \frac{2 (\frac{7 x}{2 (3 - 2 x)})}{\frac{- 8 + 15}{3 - 2 x}}$

Step 5.3.4.5.9

Add $- 8$ and $15$ .

$f (\frac{5 x - 4}{2 (3 - 2 x)}) = \frac{2 (\frac{7 x}{2 (3 - 2 x)})}{\frac{7}{3 - 2 x}}$

Step 5.3.5

Combine $2$ and $\frac{7 x}{2 (3 - 2 x)}$ .

$f (\frac{5 x - 4}{2 (3 - 2 x)}) = \frac{\frac{2 (7 x)}{2 (3 - 2 x)}}{\frac{7}{3 - 2 x}}$

Step 5.3.6

Multiply $2$ by $7$ .

$f (\frac{5 x - 4}{2 (3 - 2 x)}) = \frac{\frac{14 x}{2 (3 - 2 x)}}{\frac{7}{3 - 2 x}}$

Step 5.3.7

Reduce the expression $\frac{14 x}{2 (3 - 2 x)}$ by cancelling the common factors.

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Step 5.3.7.1

Factor $2$ out of $14 x$ .

$f (\frac{5 x - 4}{2 (3 - 2 x)}) = \frac{\frac{2 (7 x)}{2 (3 - 2 x)}}{\frac{7}{3 - 2 x}}$

Step 5.3.7.2

Cancel the common factor.

$f (\frac{5 x - 4}{2 (3 - 2 x)}) = \frac{\frac{2 (7 x)}{2 (3 - 2 x)}}{\frac{7}{3 - 2 x}}$

Step 5.3.7.3

Rewrite the expression.

$f (\frac{5 x - 4}{2 (3 - 2 x)}) = \frac{\frac{7 x}{3 - 2 x}}{\frac{7}{3 - 2 x}}$

Step 5.3.8

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{5 x - 4}{2 (3 - 2 x)}) = \frac{7 x}{3 - 2 x} \cdot \frac{3 - 2 x}{7}$

Step 5.3.9

Cancel the common factor of $7$ .

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Step 5.3.9.1

Factor $7$ out of $7 x$ .

$f (\frac{5 x - 4}{2 (3 - 2 x)}) = \frac{7 (x)}{3 - 2 x} \cdot \frac{3 - 2 x}{7}$

Step 5.3.9.2

Cancel the common factor.

$f (\frac{5 x - 4}{2 (3 - 2 x)}) = \frac{7 x}{3 - 2 x} \cdot \frac{3 - 2 x}{7}$

Step 5.3.9.3

Rewrite the expression.

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

Step 5.3.10

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

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Step 5.3.10.1

Cancel the common factor.

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

Step 5.3.10.2

Rewrite the expression.

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

Step 5.4

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

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