Find the Inverse p(m)=m/5+8

$p (m) = \frac{m}{5} + 8$

Step 1

Write $p (m) = \frac{m}{5} + 8$ as an equation.

$y = \frac{m}{5} + 8$

Step 2

Interchange the variables.

$m = \frac{y}{5} + 8$

Step 3

Solve for $y$ .

Rewrite the equation as $\frac{y}{5} + 8 = m$ .

$\frac{y}{5} + 8 = m$

Subtract $8$ from both sides of the equation.

$\frac{y}{5} = m - 8$

Multiply both sides of the equation by $5$ .

$5 \frac{y}{5} = 5 (m - 8)$

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$5 \frac{y}{5} = 5 (m - 8)$

Rewrite the expression.

$y = 5 (m - 8)$

Simplify the right side.

Simplify $5 (m - 8)$ .

Apply the distributive property.

$y = 5 m + 5 \cdot - 8$

Multiply $5$ by $- 8$ .

$y = 5 m - 40$

Step 4

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

$p^{- 1} (m) = 5 m - 40$

Step 5

Verify if $p^{- 1} (m) = 5 m - 40$ is the inverse of $p (m) = \frac{m}{5} + 8$ .

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

Evaluate $p^{- 1} (p (m))$ .

Set up the composite result function.

$p^{- 1} (p (m))$

Evaluate $p^{- 1} (\frac{m}{5} + 8)$ by substituting in the value of $p$ into $p^{- 1}$ .

$p^{- 1} (\frac{m}{5} + 8) = 5 (\frac{m}{5} + 8) - 40$

Simplify each term.

Apply the distributive property.

$p^{- 1} (\frac{m}{5} + 8) = 5 (\frac{m}{5}) + 5 \cdot 8 - 40$

Cancel the common factor of $5$ .

Cancel the common factor.

$p^{- 1} (\frac{m}{5} + 8) = 5 (\frac{m}{5}) + 5 \cdot 8 - 40$

Rewrite the expression.

$p^{- 1} (\frac{m}{5} + 8) = m + 5 \cdot 8 - 40$

Multiply $5$ by $8$ .

$p^{- 1} (\frac{m}{5} + 8) = m + 40 - 40$

Combine the opposite terms in $m + 40 - 40$ .

Subtract $40$ from $40$ .

$p^{- 1} (\frac{m}{5} + 8) = m + 0$

Add $m$ and $0$ .

$p^{- 1} (\frac{m}{5} + 8) = m$

Evaluate $p (p^{- 1} (m))$ .

Set up the composite result function.

$p (p^{- 1} (m))$

Evaluate $p (5 m - 40)$ by substituting in the value of $p^{- 1}$ into $p$ .

$p (5 m - 40) = \frac{5 m - 40}{5} + 8$

Cancel the common factor of $5 m - 40$ and $5$ .

Factor $5$ out of $5 m$ .

$p (5 m - 40) = \frac{5 (m) - 40}{5} + 8$

Factor $5$ out of $- 40$ .

$p (5 m - 40) = \frac{5 (m) + 5 \cdot - 8}{5} + 8$

Factor $5$ out of $5 (m) + 5 (- 8)$ .

$p (5 m - 40) = \frac{5 (m - 8)}{5} + 8$

Cancel the common factors.

Factor $5$ out of $5$ .

$p (5 m - 40) = \frac{5 (m - 8)}{5 (1)} + 8$

Cancel the common factor.

$p (5 m - 40) = \frac{5 (m - 8)}{5 \cdot 1} + 8$

Rewrite the expression.

$p (5 m - 40) = \frac{m - 8}{1} + 8$

Divide $m - 8$ by $1$ .

$p (5 m - 40) = m - 8 + 8$

Combine the opposite terms in $m - 8 + 8$ .

Add $- 8$ and $8$ .

$p (5 m - 40) = m + 0$

Add $m$ and $0$ .

$p (5 m - 40) = m$

Since $p^{- 1} (p (m)) = m$ and $p (p^{- 1} (m)) = m$ , then $p^{- 1} (m) = 5 m - 40$ is the inverse of $p (m) = \frac{m}{5} + 8$ .

$p^{- 1} (m) = 5 m - 40$

Find the Inverse p(m)=m/5+8

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