Find the Inverse P(m)=m/4+7

$P (m) = \frac{m}{4} + 7$

Step 1

Write $P (m) = \frac{m}{4} + 7$ as an equation.

$y = \frac{m}{4} + 7$

Step 2

Interchange the variables.

$m = \frac{y}{4} + 7$

Step 3

Solve for $y$ .

Rewrite the equation as $\frac{y}{4} + 7 = m$ .

$\frac{y}{4} + 7 = m$

Subtract $7$ from both sides of the equation.

$\frac{y}{4} = m - 7$

Multiply both sides of the equation by $4$ .

$4 \frac{y}{4} = 4 (m - 7)$

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$4 \frac{y}{4} = 4 (m - 7)$

Rewrite the expression.

$y = 4 (m - 7)$

Simplify the right side.

Simplify $4 (m - 7)$ .

Apply the distributive property.

$y = 4 m + 4 \cdot - 7$

Multiply $4$ by $- 7$ .

$y = 4 m - 28$

Step 4

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

$P^{- 1} (m) = 4 m - 28$

Step 5

Verify if $P^{- 1} (m) = 4 m - 28$ is the inverse of $P (m) = \frac{m}{4} + 7$ .

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 $P^{- 1} (\frac{m}{4} + 7)$ by substituting in the value of $P$ into $P^{- 1}$ .

$P^{- 1} (\frac{m}{4} + 7) = 4 (\frac{m}{4} + 7) - 28$

Simplify each term.

Apply the distributive property.

$P^{- 1} (\frac{m}{4} + 7) = 4 (\frac{m}{4}) + 4 \cdot 7 - 28$

Cancel the common factor of $4$ .

Cancel the common factor.

$P^{- 1} (\frac{m}{4} + 7) = 4 (\frac{m}{4}) + 4 \cdot 7 - 28$

Rewrite the expression.

$P^{- 1} (\frac{m}{4} + 7) = m + 4 \cdot 7 - 28$

Multiply $4$ by $7$ .

$P^{- 1} (\frac{m}{4} + 7) = m + 28 - 28$

Combine the opposite terms in $m + 28 - 28$ .

Subtract $28$ from $28$ .

$P^{- 1} (\frac{m}{4} + 7) = m + 0$

Add $m$ and $0$ .

$P^{- 1} (\frac{m}{4} + 7) = m$

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

Set up the composite result function.

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

Evaluate $P (4 m - 28)$ by substituting in the value of $P^{- 1}$ into $P$ .

$P (4 m - 28) = \frac{4 m - 28}{4} + 7$

Cancel the common factor of $4 m - 28$ and $4$ .

Factor $4$ out of $4 m$ .

$P (4 m - 28) = \frac{4 (m) - 28}{4} + 7$

Factor $4$ out of $- 28$ .

$P (4 m - 28) = \frac{4 (m) + 4 \cdot - 7}{4} + 7$

Factor $4$ out of $4 (m) + 4 (- 7)$ .

$P (4 m - 28) = \frac{4 (m - 7)}{4} + 7$

Cancel the common factors.

Factor $4$ out of $4$ .

$P (4 m - 28) = \frac{4 (m - 7)}{4 (1)} + 7$

Cancel the common factor.

$P (4 m - 28) = \frac{4 (m - 7)}{4 \cdot 1} + 7$

Rewrite the expression.

$P (4 m - 28) = \frac{m - 7}{1} + 7$

Divide $m - 7$ by $1$ .

$P (4 m - 28) = m - 7 + 7$

Combine the opposite terms in $m - 7 + 7$ .

Add $- 7$ and $7$ .

$P (4 m - 28) = m + 0$

Add $m$ and $0$ .

$P (4 m - 28) = m$

Since $P^{- 1} (P (m)) = m$ and $P (P^{- 1} (m)) = m$ , then $P^{- 1} (m) = 4 m - 28$ is the inverse of $P (m) = \frac{m}{4} + 7$ .

$P^{- 1} (m) = 4 m - 28$

Find the Inverse P(m)=m/4+7

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