Find the Inverse a(b)=14*(b+5)/2

$a (b) = 14 \cdot \frac{b + 5}{2}$

Step 1

Write $a (b) = 14 \cdot \frac{b + 5}{2}$ as an equation.

$y = 14 \cdot \frac{b + 5}{2}$

Step 2

Interchange the variables.

$b = 14 \cdot \frac{y + 5}{2}$

Step 3

Solve for $y$ .

Rewrite the equation as $14 \cdot \frac{y + 5}{2} = b$ .

$14 \cdot \frac{y + 5}{2} = b$

Combine $14$ and $\frac{y + 5}{2}$ .

$\frac{14 (y + 5)}{2} = b$

Divide each term in $\frac{14 (y + 5)}{2} = b$ by $14$ and simplify.

Divide each term in $\frac{14 (y + 5)}{2} = b$ by $14$ .

$\frac{\frac{14 (y + 5)}{2}}{14} = \frac{b}{14}$

Simplify the left side.

Cancel the common factor of $14$ and $2$ .

Factor $2$ out of $14 (y + 5)$ .

$\frac{\frac{2 (7 (y + 5))}{2}}{14} = \frac{b}{14}$

Cancel the common factors.

Factor $2$ out of $2$ .

$\frac{\frac{2 (7 (y + 5))}{2 (1)}}{14} = \frac{b}{14}$

Cancel the common factor.

$\frac{\frac{2 (7 (y + 5))}{2 \cdot 1}}{14} = \frac{b}{14}$

Rewrite the expression.

$\frac{\frac{7 (y + 5)}{1}}{14} = \frac{b}{14}$

Divide $7 (y + 5)$ by $1$ .

$\frac{7 (y + 5)}{14} = \frac{b}{14}$

Cancel the common factors.

Factor $7$ out of $14$ .

$\frac{7 (y + 5)}{7 \cdot 2} = \frac{b}{14}$

Cancel the common factor.

$\frac{7 (y + 5)}{7 \cdot 2} = \frac{b}{14}$

Rewrite the expression.

$\frac{y + 5}{2} = \frac{b}{14}$

Multiply both sides of the equation by $2$ .

$2 \frac{y + 5}{2} = 2 \frac{b}{14}$

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$2 \frac{y + 5}{2} = 2 \frac{b}{14}$

Rewrite the expression.

$y + 5 = 2 \frac{b}{14}$

Simplify the right side.

Cancel the common factor of $2$ .

Factor $2$ out of $14$ .

$y + 5 = 2 \frac{b}{2 (7)}$

Cancel the common factor.

$y + 5 = 2 \frac{b}{2 \cdot 7}$

Rewrite the expression.

$y + 5 = \frac{b}{7}$

Subtract $5$ from both sides of the equation.

$y = \frac{b}{7} - 5$

Step 4

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

$a^{- 1} (b) = \frac{b}{7} - 5$

Step 5

Verify if $a^{- 1} (b) = \frac{b}{7} - 5$ is the inverse of $a (b) = 14 \cdot \frac{b + 5}{2}$ .

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 $a^{- 1} (14 \cdot \frac{b + 5}{2})$ by substituting in the value of $a$ into $a^{- 1}$ .

$a^{- 1} (14 \cdot \frac{b + 5}{2}) = \frac{14 \cdot \frac{b + 5}{2}}{7} - 5$

Simplify each term.

Cancel the common factor of $14$ and $7$ .

Factor $7$ out of $14 \cdot \frac{b + 5}{2}$ .

$a^{- 1} (14 \cdot \frac{b + 5}{2}) = \frac{7 (2 \cdot \frac{b + 5}{2})}{7} - 5$

Cancel the common factors.

Factor $7$ out of $7$ .

$a^{- 1} (14 \cdot \frac{b + 5}{2}) = \frac{7 (2 \cdot \frac{b + 5}{2})}{7 (1)} - 5$

Cancel the common factor.

$a^{- 1} (14 \cdot \frac{b + 5}{2}) = \frac{7 (2 \cdot \frac{b + 5}{2})}{7 \cdot 1} - 5$

Rewrite the expression.

$a^{- 1} (14 \cdot \frac{b + 5}{2}) = \frac{2 \cdot \frac{b + 5}{2}}{1} - 5$

Divide $2 \cdot \frac{b + 5}{2}$ by $1$ .

$a^{- 1} (14 \cdot \frac{b + 5}{2}) = 2 \cdot \frac{b + 5}{2} - 5$

Cancel the common factor of $2$ .

Cancel the common factor.

$a^{- 1} (14 \cdot \frac{b + 5}{2}) = 2 \cdot \frac{b + 5}{2} - 5$

Rewrite the expression.

$a^{- 1} (14 \cdot \frac{b + 5}{2}) = b + 5 - 5$

Combine the opposite terms in $b + 5 - 5$ .

Subtract $5$ from $5$ .

$a^{- 1} (14 \cdot \frac{b + 5}{2}) = b + 0$

Add $b$ and $0$ .

$a^{- 1} (14 \cdot \frac{b + 5}{2}) = b$

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

Set up the composite result function.

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

Evaluate $a (\frac{b}{7} - 5)$ by substituting in the value of $a^{- 1}$ into $a$ .

$a (\frac{b}{7} - 5) = 14 \cdot \frac{(\frac{b}{7} - 5) + 5}{2}$

Simplify the numerator.

Add $- 5$ and $5$ .

$a (\frac{b}{7} - 5) = 14 \cdot \frac{\frac{b}{7} + 0}{2}$

Add $\frac{b}{7}$ and $0$ .

$a (\frac{b}{7} - 5) = 14 \cdot \frac{\frac{b}{7}}{2}$

Simplify terms.

Cancel the common factor of $2$ .

Factor $2$ out of $14$ .

$a (\frac{b}{7} - 5) = 2 (7) \cdot \frac{\frac{b}{7}}{2}$

Cancel the common factor.

$a (\frac{b}{7} - 5) = 2 \cdot 7 \cdot \frac{\frac{b}{7}}{2}$

Rewrite the expression.

$a (\frac{b}{7} - 5) = 7 \cdot \frac{b}{7}$

$a (\frac{b}{7} - 5) = \frac{7 b}{7}$

Cancel the common factor of $7$ .

Cancel the common factor.

$a (\frac{b}{7} - 5) = \frac{7 b}{7}$

Divide $b$ by $1$ .

$a (\frac{b}{7} - 5) = b$

Since $a^{- 1} (a (b)) = b$ and $a (a^{- 1} (b)) = b$ , then $a^{- 1} (b) = \frac{b}{7} - 5$ is the inverse of $a (b) = 14 \cdot \frac{b + 5}{2}$ .

$a^{- 1} (b) = \frac{b}{7} - 5$

Find the Inverse a(b)=14*(b+5)/2

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