Find the Inverse a(b)=12*(b+9)/2

$a (b) = 12 \cdot \frac{b + 9}{2}$

Step 1

Write $a (b) = 12 \cdot \frac{b + 9}{2}$ as an equation.

$y = 12 \cdot \frac{b + 9}{2}$

Step 2

Interchange the variables.

$b = 12 \cdot \frac{y + 9}{2}$

Step 3

Solve for $y$ .

Rewrite the equation as $12 \cdot \frac{y + 9}{2} = b$ .

$12 \cdot \frac{y + 9}{2} = b$

Combine $12$ and $\frac{y + 9}{2}$ .

$\frac{12 (y + 9)}{2} = b$

Divide each term in $\frac{12 (y + 9)}{2} = b$ by $12$ and simplify.

Divide each term in $\frac{12 (y + 9)}{2} = b$ by $12$ .

$\frac{\frac{12 (y + 9)}{2}}{12} = \frac{b}{12}$

Simplify the left side.

Cancel the common factor of $12$ and $2$ .

Factor $2$ out of $12 (y + 9)$ .

$\frac{\frac{2 (6 (y + 9))}{2}}{12} = \frac{b}{12}$

Cancel the common factors.

Factor $2$ out of $2$ .

$\frac{\frac{2 (6 (y + 9))}{2 (1)}}{12} = \frac{b}{12}$

Cancel the common factor.

$\frac{\frac{2 (6 (y + 9))}{2 \cdot 1}}{12} = \frac{b}{12}$

Rewrite the expression.

$\frac{\frac{6 (y + 9)}{1}}{12} = \frac{b}{12}$

Divide $6 (y + 9)$ by $1$ .

$\frac{6 (y + 9)}{12} = \frac{b}{12}$

Cancel the common factors.

Factor $6$ out of $12$ .

$\frac{6 (y + 9)}{6 \cdot 2} = \frac{b}{12}$

Cancel the common factor.

$\frac{6 (y + 9)}{6 \cdot 2} = \frac{b}{12}$

Rewrite the expression.

$\frac{y + 9}{2} = \frac{b}{12}$

Multiply both sides of the equation by $2$ .

$2 \frac{y + 9}{2} = 2 \frac{b}{12}$

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$2 \frac{y + 9}{2} = 2 \frac{b}{12}$

Rewrite the expression.

$y + 9 = 2 \frac{b}{12}$

Simplify the right side.

Cancel the common factor of $2$ .

Factor $2$ out of $12$ .

$y + 9 = 2 \frac{b}{2 (6)}$

Cancel the common factor.

$y + 9 = 2 \frac{b}{2 \cdot 6}$

Rewrite the expression.

$y + 9 = \frac{b}{6}$

Subtract $9$ from both sides of the equation.

$y = \frac{b}{6} - 9$

Step 4

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

$a^{- 1} (b) = \frac{b}{6} - 9$

Step 5

Verify if $a^{- 1} (b) = \frac{b}{6} - 9$ is the inverse of $a (b) = 12 \cdot \frac{b + 9}{2}$ .

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

Evaluate $a^{- 1} (a (b))$ .

Set up the composite result function.

$a^{- 1} (a (b))$

Evaluate $a^{- 1} (12 \cdot \frac{b + 9}{2})$ by substituting in the value of $a$ into $a^{- 1}$ .

$a^{- 1} (12 \cdot \frac{b + 9}{2}) = \frac{12 \cdot \frac{b + 9}{2}}{6} - 9$

Simplify each term.

Cancel the common factor of $12$ and $6$ .

Factor $6$ out of $12 \cdot \frac{b + 9}{2}$ .

$a^{- 1} (12 \cdot \frac{b + 9}{2}) = \frac{6 (2 \cdot \frac{b + 9}{2})}{6} - 9$

Cancel the common factors.

Factor $6$ out of $6$ .

$a^{- 1} (12 \cdot \frac{b + 9}{2}) = \frac{6 (2 \cdot \frac{b + 9}{2})}{6 (1)} - 9$

Cancel the common factor.

$a^{- 1} (12 \cdot \frac{b + 9}{2}) = \frac{6 (2 \cdot \frac{b + 9}{2})}{6 \cdot 1} - 9$

Rewrite the expression.

$a^{- 1} (12 \cdot \frac{b + 9}{2}) = \frac{2 \cdot \frac{b + 9}{2}}{1} - 9$

Divide $2 \cdot \frac{b + 9}{2}$ by $1$ .

$a^{- 1} (12 \cdot \frac{b + 9}{2}) = 2 \cdot \frac{b + 9}{2} - 9$

Cancel the common factor of $2$ .

Cancel the common factor.

$a^{- 1} (12 \cdot \frac{b + 9}{2}) = 2 \cdot \frac{b + 9}{2} - 9$

Rewrite the expression.

$a^{- 1} (12 \cdot \frac{b + 9}{2}) = b + 9 - 9$

Combine the opposite terms in $b + 9 - 9$ .

Subtract $9$ from $9$ .

$a^{- 1} (12 \cdot \frac{b + 9}{2}) = b + 0$

Add $b$ and $0$ .

$a^{- 1} (12 \cdot \frac{b + 9}{2}) = b$

Evaluate $a (a^{- 1} (b))$ .

Set up the composite result function.

$a (a^{- 1} (b))$

Evaluate $a (\frac{b}{6} - 9)$ by substituting in the value of $a^{- 1}$ into $a$ .

$a (\frac{b}{6} - 9) = 12 \cdot \frac{(\frac{b}{6} - 9) + 9}{2}$

Simplify the numerator.

Add $- 9$ and $9$ .

$a (\frac{b}{6} - 9) = 12 \cdot \frac{\frac{b}{6} + 0}{2}$

Add $\frac{b}{6}$ and $0$ .

$a (\frac{b}{6} - 9) = 12 \cdot \frac{\frac{b}{6}}{2}$

Simplify terms.

Cancel the common factor of $2$ .

Factor $2$ out of $12$ .

$a (\frac{b}{6} - 9) = 2 (6) \cdot \frac{\frac{b}{6}}{2}$

Cancel the common factor.

$a (\frac{b}{6} - 9) = 2 \cdot 6 \cdot \frac{\frac{b}{6}}{2}$

Rewrite the expression.

$a (\frac{b}{6} - 9) = 6 \cdot \frac{b}{6}$

$a (\frac{b}{6} - 9) = \frac{6 b}{6}$

Cancel the common factor of $6$ .

Cancel the common factor.

$a (\frac{b}{6} - 9) = \frac{6 b}{6}$

Divide $b$ by $1$ .

$a (\frac{b}{6} - 9) = b$

Since $a^{- 1} (a (b)) = b$ and $a (a^{- 1} (b)) = b$ , then $a^{- 1} (b) = \frac{b}{6} - 9$ is the inverse of $a (b) = 12 \cdot \frac{b + 9}{2}$ .

$a^{- 1} (b) = \frac{b}{6} - 9$

Find the Inverse a(b)=12*(b+9)/2

Related Questions

Support LCMGCF.com by disabling your adblocker.