Algebra Examples

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

Add $2$ to both sides of the equation.

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

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.

$- y - 3, 1, 1$

Step 3.3.2

Remove parentheses.

$- y - 3, 1, 1$

Step 3.3.3

The LCM of one and any expression is the expression.

$- y - 3$

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Apply the distributive property.

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

Step 3.4.2.2

Multiply $- \frac{3}{- y - 3} (- y)$ .

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Step 3.4.2.2.1

Multiply $- 1$ by $- 1$ .

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

Step 3.4.2.2.2

Multiply $\frac{3}{- y - 3}$ by $1$ .

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

Step 3.4.2.2.3

Combine $\frac{3}{- y - 3}$ and $y$ .

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

Step 3.4.2.3

Multiply $- \frac{3}{- y - 3} \cdot - 3$ .

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Step 3.4.2.3.1

Multiply $- 3$ by $- 1$ .

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

Step 3.4.2.3.2

Combine $3$ and $\frac{3}{- y - 3}$ .

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

Step 3.4.2.3.3

Multiply $3$ by $3$ .

$\frac{3 y}{- y - 3} + \frac{9}{- y - 3} = x (- y - 3) + 2 (- y - 3)$

Step 3.4.2.4

Combine the numerators over the common denominator.

$\frac{3 y + 9}{- y - 3} = x (- y - 3) + 2 (- y - 3)$

Step 3.4.2.5

Factor $3$ out of $3 y + 9$ .

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Step 3.4.2.5.1

Factor $3$ out of $3 y$ .

$\frac{3 (y) + 9}{- y - 3} = x (- y - 3) + 2 (- y - 3)$

Step 3.4.2.5.2

Factor $3$ out of $9$ .

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

Step 3.4.2.5.3

Factor $3$ out of $3 y + 3 \cdot 3$ .

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

Step 3.4.2.6

Cancel the common factor of $y + 3$ and $- y - 3$ .

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Step 3.4.2.6.1

Factor $- 1$ out of $y$ .

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

Step 3.4.2.6.2

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

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

Step 3.4.2.6.3

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

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

Step 3.4.2.6.4

Cancel the common factor.

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

Step 3.4.2.6.5

Divide $3 \cdot (- 1)$ by $1$ .

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

Step 3.4.2.7

Multiply $3$ by $- 1$ .

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

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Simplify each term.

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Step 3.4.3.1.1

Apply the distributive property.

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

Step 3.4.3.1.2

Rewrite using the commutative property of multiplication.

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

Step 3.4.3.1.3

Move $- 3$ to the left of $x$ .

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

Step 3.4.3.1.4

Apply the distributive property.

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

Step 3.4.3.1.5

Multiply $- 1$ by $2$ .

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

Step 3.4.3.1.6

Multiply $2$ by $- 3$ .

$- 3 = - x y - 3 x - 2 y - 6$

Step 3.5

Solve the equation.

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Step 3.5.1

Rewrite the equation as $- x y - 3 x - 2 y - 6 = - 3$ .

$- x y - 3 x - 2 y - 6 = - 3$

Step 3.5.2

Move all terms not containing $y$ to the right side of the equation.

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Step 3.5.2.1

Add $3 x$ to both sides of the equation.

$- x y - 2 y - 6 = - 3 + 3 x$

Step 3.5.2.2

Add $6$ to both sides of the equation.

$- x y - 2 y = - 3 + 3 x + 6$

Step 3.5.2.3

Add $- 3$ and $6$ .

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

Step 3.5.3

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

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Step 3.5.3.1

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

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

Step 3.5.3.2

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

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

Step 3.5.3.3

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

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

Step 3.5.4

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

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Step 3.5.4.1

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

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

Step 3.5.4.2

Simplify the left side.

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Step 3.5.4.2.1

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

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Step 3.5.4.2.1.1

Cancel the common factor.

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

Step 3.5.4.2.1.2

Divide $y$ by $1$ .

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

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.

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

Step 3.5.4.3.2

Factor $3$ out of $3 x + 3$ .

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Step 3.5.4.3.2.1

Factor $3$ out of $3 x$ .

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

Step 3.5.4.3.2.2

Factor $3$ out of $3$ .

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

Step 3.5.4.3.2.3

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

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

Step 3.5.4.3.3

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

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

Step 3.5.4.3.4

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

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

Step 3.5.4.3.5

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

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

Step 3.5.4.3.6

Rewrite negatives.

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Step 3.5.4.3.6.1

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

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

Step 3.5.4.3.6.2

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

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

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

Step 5.2.3.2

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

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

Step 5.2.3.3

Combine the numerators over the common denominator.

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

Step 5.2.3.4

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

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

Step 5.2.3.5

Combine the numerators over the common denominator.

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

Step 5.2.3.6

Reorder terms.

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

Step 5.2.3.7

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

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Step 5.2.3.7.1

Apply the distributive property.

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

Step 5.2.3.7.2

Multiply $- 1$ by $- 2$ .

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

Step 5.2.3.7.3

Multiply $- 2$ by $- 3$ .

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

Step 5.2.3.7.4

Add $- x$ and $2 x$ .

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

Step 5.2.3.7.5

Subtract $3$ from $- 3$ .

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

Step 5.2.3.7.6

Add $- 6$ and $6$ .

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

Step 5.2.3.7.7

Add $x$ and $0$ .

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

Step 5.2.4

Simplify the denominator.

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Step 5.2.4.1

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

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

Step 5.2.4.2

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

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

Step 5.2.4.3

Combine the numerators over the common denominator.

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

Step 5.2.4.4

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

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

Step 5.2.4.5

Combine the numerators over the common denominator.

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

Step 5.2.4.6

Reorder terms.

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

Step 5.2.4.7

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

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Step 5.2.4.7.1

Apply the distributive property.

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

Step 5.2.4.7.2

Multiply $- 1$ by $2$ .

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

Step 5.2.4.7.3

Multiply $2$ by $- 3$ .

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

Step 5.2.4.7.4

Apply the distributive property.

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

Step 5.2.4.7.5

Multiply $- 1$ by $- 2$ .

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

Step 5.2.4.7.6

Multiply $- 2$ by $- 3$ .

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

Step 5.2.4.7.7

Add $- 2 x$ and $2 x$ .

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

Step 5.2.4.7.8

Subtract $6$ from $0$ .

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

Step 5.2.4.7.9

Subtract $3$ from $- 6$ .

$f^{- 1} (- \frac{3}{- x - 3} - 2) = - \frac{3 (\frac{x}{- x - 3})}{\frac{- 9 + 6}{- x - 3}}$

Step 5.2.4.7.10

Add $- 9$ and $6$ .

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

Step 5.2.4.8

Move the negative in front of the fraction.

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

Step 5.2.5

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

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

Step 5.2.6

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.2.7

Rewrite using the commutative property of multiplication.

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

Step 5.2.8

Cancel the common factor of $3$ .

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Step 5.2.8.1

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

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

Step 5.2.8.2

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

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

Step 5.2.8.3

Cancel the common factor.

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

Step 5.2.8.4

Rewrite the expression.

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

Step 5.2.9

Move the negative in front of the fraction.

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

Step 5.2.10

Apply the distributive property.

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

Step 5.2.11

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

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Step 5.2.11.1

Multiply $- 1$ by $- 1$ .

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

Step 5.2.11.2

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

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

Step 5.2.11.3

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

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

Step 5.2.11.4

Raise $x$ to the power of $1$ .

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

Step 5.2.11.5

Raise $x$ to the power of $1$ .

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

Step 5.2.11.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 5.2.11.7

Add $1$ and $1$ .

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

Step 5.2.12

Multiply $- \frac{x}{- x - 3} \cdot - 3$ .

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Step 5.2.12.1

Multiply $- 3$ by $- 1$ .

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

Step 5.2.12.2

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

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

Step 5.2.13

Combine the numerators over the common denominator.

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

Step 5.2.14

Factor $x$ out of $x^{2} + 3 x$ .

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Step 5.2.14.1

Factor $x$ out of $x^{2}$ .

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

Step 5.2.14.2

Factor $x$ out of $3 x$ .

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

Step 5.2.14.3

Factor $x$ out of $x \cdot x + x \cdot 3$ .

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

Step 5.2.15

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

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Step 5.2.15.1

Factor $- 1$ out of $x$ .

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

Step 5.2.15.2

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

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

Step 5.2.15.3

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

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

Step 5.2.15.4

Cancel the common factor.

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

Step 5.2.15.5

Divide $x \cdot (- 1)$ by $1$ .

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

Step 5.2.16

Simplify the expression.

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Step 5.2.16.1

Move $- 1$ to the left of $x$ .

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

Step 5.2.16.2

Rewrite $- 1 x$ as $- x$ .

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

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

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

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Cancel the common factor of $3$ and $- (- \frac{3 (x + 1)}{x + 2}) - 3$ .

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Step 5.3.3.1.1

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

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

Step 5.3.3.1.2

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

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

Step 5.3.3.1.3

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

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

Step 5.3.3.1.4

Factor $3$ out of $3$ .

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

Step 5.3.3.1.5

Cancel the common factors.

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Step 5.3.3.1.5.1

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

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

Step 5.3.3.1.5.2

Cancel the common factor.

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

Step 5.3.3.1.5.3

Rewrite the expression.

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

Step 5.3.3.2

Cancel the common factor of $1$ and $- 1$ .

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Step 5.3.3.2.1

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

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

Step 5.3.3.2.2

Move the negative in front of the fraction.

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

Step 5.3.3.3

Simplify the denominator.

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Step 5.3.3.3.1

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

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

Step 5.3.3.3.2

Combine the numerators over the common denominator.

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

Step 5.3.3.3.3

Reorder terms.

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

Step 5.3.3.3.4

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

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Step 5.3.3.3.4.1

Apply the distributive property.

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

Step 5.3.3.3.4.2

Multiply $- 1$ by $1$ .

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

Step 5.3.3.3.4.3

Subtract $x$ from $x$ .

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

Step 5.3.3.3.4.4

Add $0$ and $2$ .

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

Step 5.3.3.3.4.5

Subtract $1$ from $2$ .

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

Step 5.3.3.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.3.3.5

Multiply $x + 2$ by $1$ .

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

Step 5.3.3.6

Apply the distributive property.

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

Step 5.3.3.7

Multiply $- 1$ by $2$ .

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

Step 5.3.3.8

Apply the distributive property.

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

Step 5.3.3.9

Multiply $- - x$ .

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Step 5.3.3.9.1

Multiply $- 1$ by $- 1$ .

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

Step 5.3.3.9.2

Multiply $x$ by $1$ .

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

Step 5.3.3.10

Multiply $- 1$ by $- 2$ .

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

Step 5.3.4

Combine the opposite terms in $x + 2 - 2$ .

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Step 5.3.4.1

Subtract $2$ from $2$ .

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

Step 5.3.4.2

Add $x$ and $0$ .

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

Step 5.4

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

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