Find the Inverse y=6-x/2

$y = 6 - \frac{x}{2}$

Step 1

Interchange the variables.

$x = 6 - \frac{y}{2}$

Step 2

Solve for $y$ .

Rewrite the equation as $6 - \frac{y}{2} = x$ .

$6 - \frac{y}{2} = x$

Subtract $6$ from both sides of the equation.

$- \frac{y}{2} = x - 6$

Multiply both sides of the equation by $- 2$ .

$- 2 (- \frac{y}{2}) = - 2 (x - 6)$

Simplify both sides of the equation.

Simplify the left side.

Simplify $- 2 (- \frac{y}{2})$ .

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{y}{2}$ into the numerator.

$- 2 \frac{- y}{2} = - 2 (x - 6)$

Factor $2$ out of $- 2$ .

$2 (- 1) \frac{- y}{2} = - 2 (x - 6)$

Cancel the common factor.

$2 \cdot - 1 \frac{- y}{2} = - 2 (x - 6)$

Rewrite the expression.

$- - y = - 2 (x - 6)$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 y = - 2 (x - 6)$

Multiply $y$ by $1$ .

$y = - 2 (x - 6)$

Simplify the right side.

Simplify $- 2 (x - 6)$ .

Apply the distributive property.

$y = - 2 x - 2 \cdot - 6$

Multiply $- 2$ by $- 6$ .

$y = - 2 x + 12$

Step 3

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

$f^{- 1} (x) = - 2 x + 12$

Step 4

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

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

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

Set up the composite result function.

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

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

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

Simplify each term.

Apply the distributive property.

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

Multiply $- 2$ by $6$ .

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

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{x}{2}$ into the numerator.

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

Factor $2$ out of $- 2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $- 1$ by $- 1$ .

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

Multiply $x$ by $1$ .

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

Combine the opposite terms in $- 12 + x + 12$ .

Add $- 12$ and $12$ .

$f^{- 1} (6 - \frac{x}{2}) = x + 0$

Add $x$ and $0$ .

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

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

Set up the composite result function.

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

Evaluate $f (- 2 x + 12)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- 2 x + 12) = 6 - \frac{- 2 x + 12}{2}$

Simplify each term.

Cancel the common factor of $- 2 x + 12$ and $2$ .

Factor $2$ out of $- 2 x$ .

$f (- 2 x + 12) = 6 - \frac{2 (- x) + 12}{2}$

Factor $2$ out of $12$ .

$f (- 2 x + 12) = 6 - \frac{2 (- x) + 2 \cdot 6}{2}$

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

$f (- 2 x + 12) = 6 - \frac{2 (- x + 6)}{2}$

Cancel the common factors.

Factor $2$ out of $2$ .

$f (- 2 x + 12) = 6 - \frac{2 (- x + 6)}{2 (1)}$

Cancel the common factor.

$f (- 2 x + 12) = 6 - \frac{2 (- x + 6)}{2 \cdot 1}$

Rewrite the expression.

$f (- 2 x + 12) = 6 - \frac{- x + 6}{1}$

Divide $- x + 6$ by $1$ .

$f (- 2 x + 12) = 6 - (- x + 6)$

Apply the distributive property.

$f (- 2 x + 12) = 6 + x - 1 \cdot 6$

Multiply $- - x$ .

Multiply $- 1$ by $- 1$ .

$f (- 2 x + 12) = 6 + 1 x - 1 \cdot 6$

Multiply $x$ by $1$ .

$f (- 2 x + 12) = 6 + x - 1 \cdot 6$

Multiply $- 1$ by $6$ .

$f (- 2 x + 12) = 6 + x - 6$

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

Subtract $6$ from $6$ .

$f (- 2 x + 12) = x + 0$

Add $x$ and $0$ .

$f (- 2 x + 12) = x$

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

$f^{- 1} (x) = - 2 x + 12$

Find the Inverse y=6-x/2

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