Finite Math Examples

$f (n) = \frac{- 12 - 2 n}{3}$

Step 1

Write $f (n) = \frac{- 12 - 2 n}{3}$ as an equation.

$y = \frac{- 12 - 2 n}{3}$

Step 2

Interchange the variables.

$n = \frac{- 12 - 2 y}{3}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{- 12 - 2 y}{3} = n$ .

$\frac{- 12 - 2 y}{3} = n$

Step 3.2

Multiply both sides by $3$ .

$\frac{- 12 - 2 y}{3} \cdot 3 = n \cdot 3$

Step 3.3

Simplify.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Simplify $\frac{- 12 - 2 y}{3} \cdot 3$ .

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Step 3.3.1.1.1

Factor $2$ out of $- 12 - 2 y$ .

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Step 3.3.1.1.1.1

Factor $2$ out of $- 12$ .

$\frac{2 (- 6) - 2 y}{3} \cdot 3 = n \cdot 3$

Step 3.3.1.1.1.2

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

$\frac{2 (- 6) + 2 (- y)}{3} \cdot 3 = n \cdot 3$

Step 3.3.1.1.1.3

Factor $2$ out of $2 (- 6) + 2 (- y)$ .

$\frac{2 (- 6 - y)}{3} \cdot 3 = n \cdot 3$

Step 3.3.1.1.2

Cancel the common factor of $3$ .

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Step 3.3.1.1.2.1

Cancel the common factor.

$\frac{2 (- 6 - y)}{3} \cdot 3 = n \cdot 3$

Step 3.3.1.1.2.2

Rewrite the expression.

$2 (- 6 - y) = n \cdot 3$

Step 3.3.1.1.3

Apply the distributive property.

$2 \cdot - 6 + 2 (- y) = n \cdot 3$

Step 3.3.1.1.4

Simplify the expression.

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Step 3.3.1.1.4.1

Multiply $2$ by $- 6$ .

$- 12 + 2 (- y) = n \cdot 3$

Step 3.3.1.1.4.2

Multiply $- 1$ by $2$ .

$- 12 - 2 y = n \cdot 3$

Step 3.3.1.1.4.3

Reorder $- 12$ and $- 2 y$ .

$- 2 y - 12 = n \cdot 3$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Move $3$ to the left of $n$ .

$- 2 y - 12 = 3 n$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Add $12$ to both sides of the equation.

$- 2 y = 3 n + 12$

Step 3.4.2

Divide each term in $- 2 y = 3 n + 12$ by $- 2$ and simplify.

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Step 3.4.2.1

Divide each term in $- 2 y = 3 n + 12$ by $- 2$ .

$\frac{- 2 y}{- 2} = \frac{3 n}{- 2} + \frac{12}{- 2}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $- 2$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{- 2 y}{- 2} = \frac{3 n}{- 2} + \frac{12}{- 2}$

Step 3.4.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{3 n}{- 2} + \frac{12}{- 2}$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Simplify each term.

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Step 3.4.2.3.1.1

Move the negative in front of the fraction.

$y = - \frac{3 n}{2} + \frac{12}{- 2}$

Step 3.4.2.3.1.2

Divide $12$ by $- 2$ .

$y = - \frac{3 n}{2} - 6$

Step 4

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

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

Step 5

Verify if $f^{- 1} (n) = - \frac{3 n}{2} - 6$ is the inverse of $f (n) = \frac{- 12 - 2 n}{3}$ .

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Step 5.1

To verify the inverse, check if $f^{- 1} (f (n)) = n$ and $f (f^{- 1} (n)) = n$ .

Step 5.2

Evaluate $f^{- 1} (f (n))$ .

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Step 5.2.1

Set up the composite result function.

$f^{- 1} (f (n))$

Step 5.2.2

Evaluate $f^{- 1} (\frac{- 12 - 2 n}{3})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{- 12 - 2 n}{3}) = - \frac{3 (\frac{- 12 - 2 n}{3})}{2} - 6$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Factor $2$ out of $- 12 - 2 n$ .

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Step 5.2.3.1.1

Factor $2$ out of $- 12$ .

$f^{- 1} (\frac{- 12 - 2 n}{3}) = - \frac{3 (\frac{2 (- 6) - 2 n}{3})}{2} - 6$

Step 5.2.3.1.2

Factor $2$ out of $- 2 n$ .

$f^{- 1} (\frac{- 12 - 2 n}{3}) = - \frac{3 (\frac{2 (- 6) + 2 (- n)}{3})}{2} - 6$

Step 5.2.3.1.3

Factor $2$ out of $2 (- 6) + 2 (- n)$ .

$f^{- 1} (\frac{- 12 - 2 n}{3}) = - \frac{3 (\frac{2 (- 6 - n)}{3})}{2} - 6$

Step 5.2.3.2

Combine $3$ and $\frac{2 (- 6 - n)}{3}$ .

$f^{- 1} (\frac{- 12 - 2 n}{3}) = - \frac{\frac{3 (2 (- 6 - n))}{3}}{2} - 6$

Step 5.2.3.3

Multiply $3$ by $2$ .

$f^{- 1} (\frac{- 12 - 2 n}{3}) = - \frac{\frac{6 (- 6 - n)}{3}}{2} - 6$

Step 5.2.3.4

Reduce the expression by cancelling the common factors.

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Step 5.2.3.4.1

Reduce the expression $\frac{6 (- 6 - n)}{3}$ by cancelling the common factors.

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Step 5.2.3.4.1.1

Factor $3$ out of $6 (- 6 - n)$ .

$f^{- 1} (\frac{- 12 - 2 n}{3}) = - \frac{\frac{3 (2 (- 6 - n))}{3}}{2} - 6$

Step 5.2.3.4.1.2

Factor $3$ out of $3$ .

$f^{- 1} (\frac{- 12 - 2 n}{3}) = - \frac{\frac{3 (2 (- 6 - n))}{3 (1)}}{2} - 6$

Step 5.2.3.4.1.3

Cancel the common factor.

$f^{- 1} (\frac{- 12 - 2 n}{3}) = - \frac{\frac{3 (2 (- 6 - n))}{3 \cdot 1}}{2} - 6$

Step 5.2.3.4.1.4

Rewrite the expression.

$f^{- 1} (\frac{- 12 - 2 n}{3}) = - \frac{\frac{2 (- 6 - n)}{1}}{2} - 6$

Step 5.2.3.4.2

Divide $2 (- 6 - n)$ by $1$ .

$f^{- 1} (\frac{- 12 - 2 n}{3}) = - \frac{2 (- 6 - n)}{2} - 6$

Step 5.2.3.5

Cancel the common factor of $2$ .

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Step 5.2.3.5.1

Cancel the common factor.

$f^{- 1} (\frac{- 12 - 2 n}{3}) = - \frac{2 (- 6 - n)}{2} - 6$

Step 5.2.3.5.2

Divide $- 6 - n$ by $1$ .

$f^{- 1} (\frac{- 12 - 2 n}{3}) = - (- 6 - n) - 6$

Step 5.2.3.6

Apply the distributive property.

$f^{- 1} (\frac{- 12 - 2 n}{3}) = 6 + n - 6$

Step 5.2.3.7

Multiply $- 1$ by $- 6$ .

$f^{- 1} (\frac{- 12 - 2 n}{3}) = 6 + n - 6$

Step 5.2.3.8

Multiply $- - n$ .

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Step 5.2.3.8.1

Multiply $- 1$ by $- 1$ .

$f^{- 1} (\frac{- 12 - 2 n}{3}) = 6 + 1 n - 6$

Step 5.2.3.8.2

Multiply $n$ by $1$ .

$f^{- 1} (\frac{- 12 - 2 n}{3}) = 6 + n - 6$

Step 5.2.4

Combine the opposite terms in $6 + n - 6$ .

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Step 5.2.4.1

Subtract $6$ from $6$ .

$f^{- 1} (\frac{- 12 - 2 n}{3}) = n + 0$

Step 5.2.4.2

Add $n$ and $0$ .

$f^{- 1} (\frac{- 12 - 2 n}{3}) = n$

Step 5.3

Evaluate $f (f^{- 1} (n))$ .

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Step 5.3.1

Set up the composite result function.

$f (f^{- 1} (n))$

Step 5.3.2

Evaluate $f (- \frac{3 n}{2} - 6)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{3 n}{2} - 6) = \frac{- 12 - 2 (- \frac{3 n}{2} - 6)}{3}$

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Factor $- 2$ out of $- 12 - 2 (- \frac{3 n}{2} - 6)$ .

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Step 5.3.3.1.1

Reorder $- 12$ and $- 2 (- \frac{3 n}{2} - 6)$ .

$f (- \frac{3 n}{2} - 6) = \frac{- 2 (- \frac{3 n}{2} - 6) - 12}{3}$

Step 5.3.3.1.2

Factor $- 2$ out of $- 12$ .

$f (- \frac{3 n}{2} - 6) = \frac{- 2 (- \frac{3 n}{2} - 6) - 2 \cdot 6}{3}$

Step 5.3.3.1.3

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

$f (- \frac{3 n}{2} - 6) = \frac{- 2 (- \frac{3 n}{2} - 6 + 6)}{3}$

Step 5.3.3.2

Add $- 6$ and $6$ .

$f (- \frac{3 n}{2} - 6) = \frac{- 2 (- \frac{3 n}{2} + 0)}{3}$

Step 5.3.3.3

Add $- \frac{3 n}{2}$ and $0$ .

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

Step 5.3.3.4

Combine exponents.

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Step 5.3.3.4.1

Factor out negative.

$f (- \frac{3 n}{2} - 6) = \frac{- (- 2 \frac{3 n}{2})}{3}$

Step 5.3.3.4.2

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

$f (- \frac{3 n}{2} - 6) = \frac{- \frac{- 2 (3 n)}{2}}{3}$

Step 5.3.3.4.3

Multiply $3$ by $- 2$ .

$f (- \frac{3 n}{2} - 6) = \frac{- \frac{- 6 n}{2}}{3}$

Step 5.3.3.5

Cancel the common factor of $- 6$ and $2$ .

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Step 5.3.3.5.1

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

$f (- \frac{3 n}{2} - 6) = \frac{- \frac{2 (- 3 n)}{2}}{3}$

Step 5.3.3.5.2

Cancel the common factors.

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Step 5.3.3.5.2.1

Factor $2$ out of $2$ .

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

Step 5.3.3.5.2.2

Cancel the common factor.

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

Step 5.3.3.5.2.3

Rewrite the expression.

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

Step 5.3.3.5.2.4

Divide $- 3 n$ by $1$ .

$f (- \frac{3 n}{2} - 6) = \frac{- (- 3 n)}{3}$

$f (- \frac{3 n}{2} - 6) = \frac{3 n}{3}$

Step 5.3.3.6

Multiply $- 1$ by $- 3$ .

$f (- \frac{3 n}{2} - 6) = \frac{3 n}{3}$

Step 5.3.4

Cancel the common factor of $3$ .

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Step 5.3.4.1

Cancel the common factor.

$f (- \frac{3 n}{2} - 6) = \frac{3 n}{3}$

Step 5.3.4.2

Divide $n$ by $1$ .

$f (- \frac{3 n}{2} - 6) = n$

Step 5.4

Since $f^{- 1} (f (n)) = n$ and $f (f^{- 1} (n)) = n$ , then $f^{- 1} (n) = - \frac{3 n}{2} - 6$ is the inverse of $f (n) = \frac{- 12 - 2 n}{3}$ .

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