Precalculus Examples

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

Step 1

Write $f (x) = \frac{4 x - 2}{3 x + 1}$ as an equation.

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{4 y - 2}{3 y + 1} = x$ .

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

Step 3.2

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

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Step 3.2.1

Factor $2$ out of $4 y$ .

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

Step 3.2.2

Factor $2$ out of $- 2$ .

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

Step 3.2.3

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

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

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.

$3 y + 1, 1$

Step 3.3.2

Remove parentheses.

$3 y + 1, 1$

Step 3.3.3

The LCM of one and any expression is the expression.

$3 y + 1$

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $3 y + 1$ .

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Step 3.4.2.1.1

Cancel the common factor.

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

Step 3.4.2.1.2

Rewrite the expression.

$2 (2 y - 1) = x (3 y + 1)$

Step 3.4.2.2

Apply the distributive property.

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

Step 3.4.2.3

Multiply.

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Step 3.4.2.3.1

Multiply $2$ by $2$ .

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

Step 3.4.2.3.2

Multiply $2$ by $- 1$ .

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

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Apply the distributive property.

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

Step 3.4.3.2

Simplify the expression.

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Step 3.4.3.2.1

Rewrite using the commutative property of multiplication.

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

Step 3.4.3.2.2

Multiply $x$ by $1$ .

$4 y - 2 = 3 x y + x$

Step 3.5

Solve the equation.

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Step 3.5.1

Subtract $3 x y$ from both sides of the equation.

$4 y - 2 - 3 x y = x$

Step 3.5.2

Add $2$ to both sides of the equation.

$4 y - 3 x y = x + 2$

Step 3.5.3

Factor $y$ out of $4 y - 3 x y$ .

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Step 3.5.3.1

Factor $y$ out of $4 y$ .

$y \cdot 4 - 3 x y = x + 2$

Step 3.5.3.2

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

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

Step 3.5.3.3

Factor $y$ out of $y \cdot 4 + y (- 3 x)$ .

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

Step 3.5.4

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

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Step 3.5.4.1

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

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

Step 3.5.4.2

Simplify the left side.

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Step 3.5.4.2.1

Cancel the common factor of $4 - 3 x$ .

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Step 3.5.4.2.1.1

Cancel the common factor.

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

Step 3.5.4.2.1.2

Divide $y$ by $1$ .

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

Step 3.5.4.3

Simplify the right side.

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Step 3.5.4.3.1

Combine the numerators over the common denominator.

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

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Multiply the numerator and denominator of the fraction by $3 x + 1$ .

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Step 5.2.3.1

Multiply $\frac{\frac{4 x - 2}{3 x + 1} + 2}{4 - 3 \frac{4 x - 2}{3 x + 1}}$ by $\frac{3 x + 1}{3 x + 1}$ .

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

Step 5.2.3.2

Combine.

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

Step 5.2.4

Apply the distributive property.

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

Step 5.2.5

Simplify with factoring out.

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Step 5.2.5.1

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

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Step 5.2.5.1.1

Factor $2$ out of $4 x$ .

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

Step 5.2.5.1.2

Factor $2$ out of $- 2$ .

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

Step 5.2.5.1.3

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

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

Step 5.2.5.2

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

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Step 5.2.5.2.1

Factor $2$ out of $4 x$ .

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

Step 5.2.5.2.2

Factor $2$ out of $- 2$ .

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

Step 5.2.5.2.3

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

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

Step 5.2.6

Simplify the numerator.

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Step 5.2.6.1

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

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Step 5.2.6.1.1

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

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

Step 5.2.6.1.2

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

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

Step 5.2.6.1.3

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

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

Step 5.2.6.2

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

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

Step 5.2.6.3

Combine the numerators over the common denominator.

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

Step 5.2.6.4

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

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Step 5.2.6.4.1

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

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

Step 5.2.6.4.2

Add $2 x$ and $3 x$ .

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

Step 5.2.6.4.3

Add $- 1$ and $1$ .

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

Step 5.2.6.4.4

Add $5 x$ and $0$ .

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

Step 5.2.6.4.5

Multiply $2$ by $5$ .

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

Step 5.2.7

Simplify the denominator.

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Step 5.2.7.1

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

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Step 5.2.7.1.1

Factor $3 x + 1$ out of $(3 x + 1) \cdot 4$ .

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

Step 5.2.7.1.2

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

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

Step 5.2.7.2

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

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Step 5.2.7.2.1

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

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

Step 5.2.7.2.2

Multiply $2$ by $- 3$ .

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

Step 5.2.7.3

Move the negative in front of the fraction.

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

Step 5.2.7.4

To write $4$ as a fraction with a common denominator, multiply by $\frac{3 x + 1}{3 x + 1}$ .

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

Step 5.2.7.5

Combine the numerators over the common denominator.

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

Step 5.2.7.6

Rewrite $\frac{4 (3 x + 1) - 6 (2 x - 1)}{3 x + 1}$ in a factored form.

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Step 5.2.7.6.1

Factor $2$ out of $4 (3 x + 1) - 6 (2 x - 1)$ .

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Step 5.2.7.6.1.1

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

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

Step 5.2.7.6.1.2

Factor $2$ out of $- 6 (2 x - 1)$ .

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

Step 5.2.7.6.1.3

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

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

Step 5.2.7.6.2

Apply the distributive property.

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

Step 5.2.7.6.3

Multiply $3$ by $2$ .

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

Step 5.2.7.6.4

Multiply $2$ by $1$ .

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

Step 5.2.7.6.5

Apply the distributive property.

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

Step 5.2.7.6.6

Multiply $2$ by $- 3$ .

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

Step 5.2.7.6.7

Multiply $- 3$ by $- 1$ .

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

Step 5.2.7.6.8

Subtract $6 x$ from $6 x$ .

$f^{- 1} (\frac{4 x - 2}{3 x + 1}) = \frac{(3 x + 1) (\frac{10 x}{3 x + 1})}{(3 x + 1) (\frac{2 (0 + 2 + 3)}{3 x + 1})}$

Step 5.2.7.6.9

Add $0$ and $2$ .

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

Step 5.2.7.6.10

Add $2$ and $3$ .

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

Step 5.2.7.7

Multiply $2$ by $5$ .

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

Step 5.2.8

Cancel the common factor of $3 x + 1$ .

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Step 5.2.8.1

Cancel the common factor.

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

Step 5.2.8.2

Rewrite the expression.

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

Step 5.2.9

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.2.10

Cancel the common factor of $10$ .

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Step 5.2.10.1

Factor $10$ out of $10 x$ .

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

Step 5.2.10.2

Cancel the common factor.

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

Step 5.2.10.3

Rewrite the expression.

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

Step 5.2.11

Cancel the common factor of $3 x + 1$ .

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Step 5.2.11.1

Cancel the common factor.

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

Step 5.2.11.2

Rewrite the expression.

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

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

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

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

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

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Step 5.3.3.1.1

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

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

Step 5.3.3.1.2

Factor $2$ out of $- 2$ .

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

Step 5.3.3.1.3

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

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

Step 5.3.3.2

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

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

Step 5.3.3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4 - 3 x}{4 - 3 x}$ .

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

Step 5.3.3.4

Combine $- 1$ and $\frac{4 - 3 x}{4 - 3 x}$ .

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

Step 5.3.3.5

Combine the numerators over the common denominator.

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

Step 5.3.3.6

Reorder terms.

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

Step 5.3.3.7

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

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Step 5.3.3.7.1

Apply the distributive property.

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

Step 5.3.3.7.2

Multiply $2$ by $2$ .

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

Step 5.3.3.7.3

Apply the distributive property.

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

Step 5.3.3.7.4

Multiply $- 1$ by $4$ .

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

Step 5.3.3.7.5

Multiply $- 3$ by $- 1$ .

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

Step 5.3.3.7.6

Add $2 x$ and $3 x$ .

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

Step 5.3.3.7.7

Subtract $4$ from $4$ .

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

Step 5.3.3.7.8

Add $5 x$ and $0$ .

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

Step 5.3.4

Simplify the denominator.

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Step 5.3.4.1

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

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

Step 5.3.4.2

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

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

Step 5.3.4.3

Combine the numerators over the common denominator.

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

Step 5.3.4.4

Reorder terms.

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

Step 5.3.4.5

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

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Step 5.3.4.5.1

Apply the distributive property.

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

Step 5.3.4.5.2

Multiply $3$ by $2$ .

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

Step 5.3.4.5.3

Add $- 3 x$ and $3 x$ .

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

Step 5.3.4.5.4

Add $0$ and $6$ .

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

Step 5.3.4.5.5

Add $6$ and $4$ .

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

Step 5.3.5

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

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

Step 5.3.6

Multiply $2$ by $5$ .

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

Step 5.3.7

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.3.8

Cancel the common factor of $10$ .

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Step 5.3.8.1

Factor $10$ out of $10 x$ .

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

Step 5.3.8.2

Cancel the common factor.

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

Step 5.3.8.3

Rewrite the expression.

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

Step 5.3.9

Multiply $\frac{x}{- 3 x + 4}$ by $- 3 x + 4$ .

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

Step 5.3.10

Cancel the common factor of $- 3 x + 4$ .

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Step 5.3.10.1

Cancel the common factor.

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

Step 5.3.10.2

Divide $x$ by $1$ .

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

Step 5.4

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

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