Finite Math Examples

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

Step 1

Write $f (x) = \frac{- 3 x + 2}{7 x + 2}$ as an equation.

$y = \frac{- 3 x + 2}{7 x + 2}$

Step 2

Interchange the variables.

$x = \frac{- 3 y + 2}{7 y + 2}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{- 3 y + 2}{7 y + 2} = x$ .

$\frac{- 3 y + 2}{7 y + 2} = x$

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.

$7 y + 2, 1$

Step 3.2.2

Remove parentheses.

$7 y + 2, 1$

Step 3.2.3

The LCM of one and any expression is the expression.

$7 y + 2$

Step 3.3

Multiply each term in $\frac{- 3 y + 2}{7 y + 2} = x$ by $7 y + 2$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $\frac{- 3 y + 2}{7 y + 2} = x$ by $7 y + 2$ .

$\frac{- 3 y + 2}{7 y + 2} (7 y + 2) = x (7 y + 2)$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $7 y + 2$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{- 3 y + 2}{7 y + 2} (7 y + 2) = x (7 y + 2)$

Step 3.3.2.1.2

Rewrite the expression.

$- 3 y + 2 = x (7 y + 2)$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Apply the distributive property.

$- 3 y + 2 = x (7 y) + x \cdot 2$

Step 3.3.3.2

Reorder.

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Step 3.3.3.2.1

Rewrite using the commutative property of multiplication.

$- 3 y + 2 = 7 x y + x \cdot 2$

Step 3.3.3.2.2

Move $2$ to the left of $x$ .

$- 3 y + 2 = 7 x y + 2 x$

Step 3.4

Solve the equation.

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Step 3.4.1

Subtract $7 x y$ from both sides of the equation.

$- 3 y + 2 - 7 x y = 2 x$

Step 3.4.2

Subtract $2$ from both sides of the equation.

$- 3 y - 7 x y = 2 x - 2$

Step 3.4.3

Factor $y$ out of $- 3 y - 7 x y$ .

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Step 3.4.3.1

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

$y \cdot - 3 - 7 x y = 2 x - 2$

Step 3.4.3.2

Factor $y$ out of $- 7 x y$ .

$y \cdot - 3 + y (- 7 x) = 2 x - 2$

Step 3.4.3.3

Factor $y$ out of $y \cdot - 3 + y (- 7 x)$ .

$y (- 3 - 7 x) = 2 x - 2$

Step 3.4.4

Divide each term in $y (- 3 - 7 x) = 2 x - 2$ by $- 3 - 7 x$ and simplify.

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Step 3.4.4.1

Divide each term in $y (- 3 - 7 x) = 2 x - 2$ by $- 3 - 7 x$ .

$\frac{y (- 3 - 7 x)}{- 3 - 7 x} = \frac{2 x}{- 3 - 7 x} + \frac{- 2}{- 3 - 7 x}$

Step 3.4.4.2

Simplify the left side.

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Step 3.4.4.2.1

Cancel the common factor of $- 3 - 7 x$ .

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Step 3.4.4.2.1.1

Cancel the common factor.

$\frac{y (- 3 - 7 x)}{- 3 - 7 x} = \frac{2 x}{- 3 - 7 x} + \frac{- 2}{- 3 - 7 x}$

Step 3.4.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{2 x}{- 3 - 7 x} + \frac{- 2}{- 3 - 7 x}$

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.

$y = \frac{2 x - 2}{- 3 - 7 x}$

Step 3.4.4.3.2

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

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Step 3.4.4.3.2.1

Factor $2$ out of $2 x$ .

$y = \frac{2 (x) - 2}{- 3 - 7 x}$

Step 3.4.4.3.2.2

Factor $2$ out of $- 2$ .

$y = \frac{2 (x) + 2 (- 1)}{- 3 - 7 x}$

Step 3.4.4.3.2.3

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

$y = \frac{2 (x - 1)}{- 3 - 7 x}$

Step 3.4.4.3.3

Rewrite $- 3$ as $- 1 (3)$ .

$y = \frac{2 (x - 1)}{- 1 (3) - 7 x}$

Step 3.4.4.3.4

Factor $- 1$ out of $- 7 x$ .

$y = \frac{2 (x - 1)}{- 1 (3) - (7 x)}$

Step 3.4.4.3.5

Factor $- 1$ out of $- 1 (3) - (7 x)$ .

$y = \frac{2 (x - 1)}{- 1 (3 + 7 x)}$

Step 3.4.4.3.6

Move the negative in front of the fraction.

$y = - \frac{2 (x - 1)}{3 + 7 x}$

Step 4

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

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

Step 5

Verify if $f^{- 1} (x) = - \frac{2 (x - 1)}{3 + 7 x}$ is the inverse of $f (x) = \frac{- 3 x + 2}{7 x + 2}$ .

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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 $f^{- 1} (\frac{- 3 x + 2}{7 x + 2})$ by substituting in the value of $f$ into $f^{- 1}$ .

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

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{7 x + 2}{7 x + 2}$ .

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

Step 5.2.3.2

Combine $- 1$ and $\frac{7 x + 2}{7 x + 2}$ .

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

Step 5.2.3.3

Combine the numerators over the common denominator.

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

Step 5.2.3.4

Rewrite $\frac{- 3 x + 2 - (7 x + 2)}{7 x + 2}$ in a factored form.

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Step 5.2.3.4.1

Apply the distributive property.

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

Step 5.2.3.4.2

Multiply $7$ by $- 1$ .

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

Step 5.2.3.4.3

Multiply $- 1$ by $2$ .

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

Step 5.2.3.4.4

Subtract $7 x$ from $- 3 x$ .

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

Step 5.2.3.4.5

Subtract $2$ from $2$ .

$f^{- 1} (\frac{- 3 x + 2}{7 x + 2}) = - \frac{2 (\frac{- 10 x + 0}{7 x + 2})}{3 + 7 (\frac{- 3 x + 2}{7 x + 2})}$

Step 5.2.3.4.6

Add $- 10 x$ and $0$ .

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

Step 5.2.3.5

Move the negative in front of the fraction.

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

Step 5.2.3.6

Combine exponents.

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Step 5.2.3.6.1

Factor out negative.

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

Step 5.2.3.6.2

Combine $2$ and $\frac{10 x}{7 x + 2}$ .

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

Step 5.2.3.6.3

Multiply $10$ by $2$ .

$f^{- 1} (\frac{- 3 x + 2}{7 x + 2}) = - \frac{- \frac{20 x}{7 x + 2}}{3 + 7 (\frac{- 3 x + 2}{7 x + 2})}$

Step 5.2.4

Simplify the denominator.

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Step 5.2.4.1

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

$f^{- 1} (\frac{- 3 x + 2}{7 x + 2}) = - \frac{- \frac{20 x}{7 x + 2}}{3 + \frac{7 (- 3 x + 2)}{7 x + 2}}$

Step 5.2.4.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{7 x + 2}{7 x + 2}$ .

$f^{- 1} (\frac{- 3 x + 2}{7 x + 2}) = - \frac{- \frac{20 x}{7 x + 2}}{\frac{3 (7 x + 2)}{7 x + 2} + \frac{7 (- 3 x + 2)}{7 x + 2}}$

Step 5.2.4.3

Combine the numerators over the common denominator.

$f^{- 1} (\frac{- 3 x + 2}{7 x + 2}) = - \frac{- \frac{20 x}{7 x + 2}}{\frac{3 (7 x + 2) + 7 (- 3 x + 2)}{7 x + 2}}$

Step 5.2.4.4

Reorder terms.

$f^{- 1} (\frac{- 3 x + 2}{7 x + 2}) = - \frac{- \frac{20 x}{7 x + 2}}{\frac{7 (- 3 x + 2) + 3 (7 x + 2)}{7 x + 2}}$

Step 5.2.4.5

Rewrite $\frac{7 (- 3 x + 2) + 3 (7 x + 2)}{7 x + 2}$ in a factored form.

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Step 5.2.4.5.1

Apply the distributive property.

$f^{- 1} (\frac{- 3 x + 2}{7 x + 2}) = - \frac{- \frac{20 x}{7 x + 2}}{\frac{7 (- 3 x) + 7 \cdot 2 + 3 (7 x + 2)}{7 x + 2}}$

Step 5.2.4.5.2

Multiply $- 3$ by $7$ .

$f^{- 1} (\frac{- 3 x + 2}{7 x + 2}) = - \frac{- \frac{20 x}{7 x + 2}}{\frac{- 21 x + 7 \cdot 2 + 3 (7 x + 2)}{7 x + 2}}$

Step 5.2.4.5.3

Multiply $7$ by $2$ .

$f^{- 1} (\frac{- 3 x + 2}{7 x + 2}) = - \frac{- \frac{20 x}{7 x + 2}}{\frac{- 21 x + 14 + 3 (7 x + 2)}{7 x + 2}}$

Step 5.2.4.5.4

Apply the distributive property.

$f^{- 1} (\frac{- 3 x + 2}{7 x + 2}) = - \frac{- \frac{20 x}{7 x + 2}}{\frac{- 21 x + 14 + 3 (7 x) + 3 \cdot 2}{7 x + 2}}$

Step 5.2.4.5.5

Multiply $7$ by $3$ .

$f^{- 1} (\frac{- 3 x + 2}{7 x + 2}) = - \frac{- \frac{20 x}{7 x + 2}}{\frac{- 21 x + 14 + 21 x + 3 \cdot 2}{7 x + 2}}$

Step 5.2.4.5.6

Multiply $3$ by $2$ .

$f^{- 1} (\frac{- 3 x + 2}{7 x + 2}) = - \frac{- \frac{20 x}{7 x + 2}}{\frac{- 21 x + 14 + 21 x + 6}{7 x + 2}}$

Step 5.2.4.5.7

Add $- 21 x$ and $21 x$ .

$f^{- 1} (\frac{- 3 x + 2}{7 x + 2}) = - \frac{- \frac{20 x}{7 x + 2}}{\frac{0 + 14 + 6}{7 x + 2}}$

Step 5.2.4.5.8

Add $0$ and $14$ .

$f^{- 1} (\frac{- 3 x + 2}{7 x + 2}) = - \frac{- \frac{20 x}{7 x + 2}}{\frac{14 + 6}{7 x + 2}}$

Step 5.2.4.5.9

Add $14$ and $6$ .

$f^{- 1} (\frac{- 3 x + 2}{7 x + 2}) = - \frac{- \frac{20 x}{7 x + 2}}{\frac{20}{7 x + 2}}$

Step 5.2.5

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{- 3 x + 2}{7 x + 2}) = - (- \frac{20 x}{7 x + 2} \cdot \frac{7 x + 2}{20})$

Step 5.2.6

Cancel the common factor of $20$ .

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Step 5.2.6.1

Move the leading negative in $- \frac{20 x}{7 x + 2}$ into the numerator.

$f^{- 1} (\frac{- 3 x + 2}{7 x + 2}) = - (\frac{- 20 x}{7 x + 2} \cdot \frac{7 x + 2}{20})$

Step 5.2.6.2

Factor $20$ out of $- 20 x$ .

$f^{- 1} (\frac{- 3 x + 2}{7 x + 2}) = - (\frac{20 (- x)}{7 x + 2} \cdot \frac{7 x + 2}{20})$

Step 5.2.6.3

Cancel the common factor.

$f^{- 1} (\frac{- 3 x + 2}{7 x + 2}) = - (\frac{20 (- x)}{7 x + 2} \cdot \frac{7 x + 2}{20})$

Step 5.2.6.4

Rewrite the expression.

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

Step 5.2.7

Cancel the common factor of $7 x + 2$ .

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Step 5.2.7.1

Cancel the common factor.

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

Step 5.2.7.2

Rewrite the expression.

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

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 $f (- \frac{2 (x - 1)}{3 + 7 x})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{2 (x - 1)}{3 + 7 x}) = \frac{- 3 (- \frac{2 (x - 1)}{3 + 7 x}) + 2}{7 (- \frac{2 (x - 1)}{3 + 7 x}) + 2}$

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Multiply $- 3 (- \frac{2 (x - 1)}{3 + 7 x})$ .

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Step 5.3.3.1.1

Multiply $- 1$ by $- 3$ .

$f (- \frac{2 (x - 1)}{3 + 7 x}) = \frac{3 (\frac{2 (x - 1)}{3 + 7 x}) + 2}{7 (- \frac{2 (x - 1)}{3 + 7 x}) + 2}$

Step 5.3.3.1.2

Combine $3$ and $\frac{2 (x - 1)}{3 + 7 x}$ .

$f (- \frac{2 (x - 1)}{3 + 7 x}) = \frac{\frac{3 (2 (x - 1))}{3 + 7 x} + 2}{7 (- \frac{2 (x - 1)}{3 + 7 x}) + 2}$

Step 5.3.3.1.3

Multiply $2$ by $3$ .

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

Step 5.3.3.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{3 + 7 x}{3 + 7 x}$ .

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

Step 5.3.3.3

Combine the numerators over the common denominator.

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

Step 5.3.3.4

Reorder terms.

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

Step 5.3.3.5

Rewrite $\frac{6 (x - 1) + 2 (3 + 7 x)}{7 x + 3}$ in a factored form.

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Step 5.3.3.5.1

Factor $2$ out of $6 (x - 1) + 2 (3 + 7 x)$ .

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Step 5.3.3.5.1.1

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

$f (- \frac{2 (x - 1)}{3 + 7 x}) = \frac{\frac{2 (3 (x - 1)) + 2 (3 + 7 x)}{7 x + 3}}{7 (- \frac{2 (x - 1)}{3 + 7 x}) + 2}$

Step 5.3.3.5.1.2

Factor $2$ out of $2 (3 (x - 1)) + 2 (3 + 7 x)$ .

$f (- \frac{2 (x - 1)}{3 + 7 x}) = \frac{\frac{2 (3 (x - 1) + 3 + 7 x)}{7 x + 3}}{7 (- \frac{2 (x - 1)}{3 + 7 x}) + 2}$

Step 5.3.3.5.2

Apply the distributive property.

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

Step 5.3.3.5.3

Multiply $3$ by $- 1$ .

$f (- \frac{2 (x - 1)}{3 + 7 x}) = \frac{\frac{2 (3 x - 3 + 3 + 7 x)}{7 x + 3}}{7 (- \frac{2 (x - 1)}{3 + 7 x}) + 2}$

Step 5.3.3.5.4

Add $3 x$ and $7 x$ .

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

Step 5.3.3.5.5

Add $- 3$ and $3$ .

$f (- \frac{2 (x - 1)}{3 + 7 x}) = \frac{\frac{2 (10 x + 0)}{7 x + 3}}{7 (- \frac{2 (x - 1)}{3 + 7 x}) + 2}$

Step 5.3.3.5.6

Add $10 x$ and $0$ .

$f (- \frac{2 (x - 1)}{3 + 7 x}) = \frac{\frac{2 \cdot (10 x)}{7 x + 3}}{7 (- \frac{2 (x - 1)}{3 + 7 x}) + 2}$

Step 5.3.3.5.7

Multiply $2$ by $10$ .

$f (- \frac{2 (x - 1)}{3 + 7 x}) = \frac{\frac{20 x}{7 x + 3}}{7 (- \frac{2 (x - 1)}{3 + 7 x}) + 2}$

Step 5.3.4

Simplify the denominator.

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Step 5.3.4.1

Multiply $7 (- \frac{2 (x - 1)}{3 + 7 x})$ .

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Step 5.3.4.1.1

Multiply $- 1$ by $7$ .

$f (- \frac{2 (x - 1)}{3 + 7 x}) = \frac{\frac{20 x}{7 x + 3}}{- 7 \frac{2 (x - 1)}{3 + 7 x} + 2}$

Step 5.3.4.1.2

Combine $- 7$ and $\frac{2 (x - 1)}{3 + 7 x}$ .

$f (- \frac{2 (x - 1)}{3 + 7 x}) = \frac{\frac{20 x}{7 x + 3}}{\frac{- 7 (2 (x - 1))}{3 + 7 x} + 2}$

Step 5.3.4.1.3

Multiply $2$ by $- 7$ .

$f (- \frac{2 (x - 1)}{3 + 7 x}) = \frac{\frac{20 x}{7 x + 3}}{\frac{- 14 (x - 1)}{3 + 7 x} + 2}$

Step 5.3.4.2

Move the negative in front of the fraction.

$f (- \frac{2 (x - 1)}{3 + 7 x}) = \frac{\frac{20 x}{7 x + 3}}{- \frac{14 (x - 1)}{3 + 7 x} + 2}$

Step 5.3.4.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{3 + 7 x}{3 + 7 x}$ .

$f (- \frac{2 (x - 1)}{3 + 7 x}) = \frac{\frac{20 x}{7 x + 3}}{- \frac{14 (x - 1)}{3 + 7 x} + \frac{2 (3 + 7 x)}{3 + 7 x}}$

Step 5.3.4.4

Combine the numerators over the common denominator.

$f (- \frac{2 (x - 1)}{3 + 7 x}) = \frac{\frac{20 x}{7 x + 3}}{\frac{- 14 (x - 1) + 2 (3 + 7 x)}{3 + 7 x}}$

Step 5.3.4.5

Rewrite $\frac{- 14 (x - 1) + 2 (3 + 7 x)}{3 + 7 x}$ in a factored form.

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Step 5.3.4.5.1

Factor $2$ out of $- 14 (x - 1) + 2 (3 + 7 x)$ .

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Step 5.3.4.5.1.1

Factor $2$ out of $- 14 (x - 1)$ .

$f (- \frac{2 (x - 1)}{3 + 7 x}) = \frac{\frac{20 x}{7 x + 3}}{\frac{2 (- 7 (x - 1)) + 2 (3 + 7 x)}{3 + 7 x}}$

Step 5.3.4.5.1.2

Factor $2$ out of $2 (- 7 (x - 1)) + 2 (3 + 7 x)$ .

$f (- \frac{2 (x - 1)}{3 + 7 x}) = \frac{\frac{20 x}{7 x + 3}}{\frac{2 (- 7 (x - 1) + 3 + 7 x)}{3 + 7 x}}$

Step 5.3.4.5.2

Apply the distributive property.

$f (- \frac{2 (x - 1)}{3 + 7 x}) = \frac{\frac{20 x}{7 x + 3}}{\frac{2 (- 7 x - 7 \cdot - 1 + 3 + 7 x)}{3 + 7 x}}$

Step 5.3.4.5.3

Multiply $- 7$ by $- 1$ .

$f (- \frac{2 (x - 1)}{3 + 7 x}) = \frac{\frac{20 x}{7 x + 3}}{\frac{2 (- 7 x + 7 + 3 + 7 x)}{3 + 7 x}}$

Step 5.3.4.5.4

Add $- 7 x$ and $7 x$ .

$f (- \frac{2 (x - 1)}{3 + 7 x}) = \frac{\frac{20 x}{7 x + 3}}{\frac{2 (0 + 7 + 3)}{3 + 7 x}}$

Step 5.3.4.5.5

Add $0$ and $7$ .

$f (- \frac{2 (x - 1)}{3 + 7 x}) = \frac{\frac{20 x}{7 x + 3}}{\frac{2 (7 + 3)}{3 + 7 x}}$

Step 5.3.4.5.6

Add $7$ and $3$ .

$f (- \frac{2 (x - 1)}{3 + 7 x}) = \frac{\frac{20 x}{7 x + 3}}{\frac{2 \cdot 10}{3 + 7 x}}$

Step 5.3.4.6

Multiply $2$ by $10$ .

$f (- \frac{2 (x - 1)}{3 + 7 x}) = \frac{\frac{20 x}{7 x + 3}}{\frac{20}{3 + 7 x}}$

Step 5.3.5

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{2 (x - 1)}{3 + 7 x}) = \frac{20 x}{7 x + 3} \cdot \frac{3 + 7 x}{20}$

Step 5.3.6

Cancel the common factor of $20$ .

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Step 5.3.6.1

Factor $20$ out of $20 x$ .

$f (- \frac{2 (x - 1)}{3 + 7 x}) = \frac{20 (x)}{7 x + 3} \cdot \frac{3 + 7 x}{20}$

Step 5.3.6.2

Cancel the common factor.

$f (- \frac{2 (x - 1)}{3 + 7 x}) = \frac{20 x}{7 x + 3} \cdot \frac{3 + 7 x}{20}$

Step 5.3.6.3

Rewrite the expression.

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

Step 5.3.7

Multiply $\frac{x}{7 x + 3}$ by $3 + 7 x$ .

$f (- \frac{2 (x - 1)}{3 + 7 x}) = \frac{x (3 + 7 x)}{7 x + 3}$

Step 5.3.8

Cancel the common factor of $3 + 7 x$ and $7 x + 3$ .

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Step 5.3.8.1

Reorder terms.

$f (- \frac{2 (x - 1)}{3 + 7 x}) = \frac{x (7 x + 3)}{7 x + 3}$

Step 5.3.8.2

Cancel the common factor.

$f (- \frac{2 (x - 1)}{3 + 7 x}) = \frac{x (7 x + 3)}{7 x + 3}$

Step 5.3.8.3

Divide $x$ by $1$ .

$f (- \frac{2 (x - 1)}{3 + 7 x}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{2 (x - 1)}{3 + 7 x}$ is the inverse of $f (x) = \frac{- 3 x + 2}{7 x + 2}$ .

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