Find the Inverse x/2-12

$\frac{x}{2} - 12$

Step 1

Interchange the variables.

$x = \frac{y}{2} - 12$

Step 2

Solve for $y$ .

Rewrite the equation as $\frac{y}{2} - 12 = x$ .

$\frac{y}{2} - 12 = x$

Add $12$ to both sides of the equation.

$\frac{y}{2} = x + 12$

Multiply both sides of the equation by $2$ .

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

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$y = 2 (x + 12)$

Simplify the right side.

Simplify $2 (x + 12)$ .

Apply the distributive property.

$y = 2 x + 2 \cdot 12$

Multiply $2$ by $12$ .

$y = 2 x + 24$

Step 3

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

$f^{- 1} (x) = 2 x + 24$

Step 4

Verify if $f^{- 1} (x) = 2 x + 24$ is the inverse of $f (x) = \frac{x}{2} - 12$ .

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

$f^{- 1} (\frac{x}{2} - 12) = 2 (\frac{x}{2} - 12) + 24$

Simplify each term.

Apply the distributive property.

$f^{- 1} (\frac{x}{2} - 12) = 2 (\frac{x}{2}) + 2 \cdot - 12 + 24$

Cancel the common factor of $2$ .

Cancel the common factor.

$f^{- 1} (\frac{x}{2} - 12) = 2 (\frac{x}{2}) + 2 \cdot - 12 + 24$

Rewrite the expression.

$f^{- 1} (\frac{x}{2} - 12) = x + 2 \cdot - 12 + 24$

Multiply $2$ by $- 12$ .

$f^{- 1} (\frac{x}{2} - 12) = x - 24 + 24$

Combine the opposite terms in $x - 24 + 24$ .

Add $- 24$ and $24$ .

$f^{- 1} (\frac{x}{2} - 12) = x + 0$

Add $x$ and $0$ .

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

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

Set up the composite result function.

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

Evaluate $f (2 x + 24)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (2 x + 24) = \frac{2 x + 24}{2} - 12$

Cancel the common factor of $2 x + 24$ and $2$ .

Factor $2$ out of $2 x$ .

$f (2 x + 24) = \frac{2 (x) + 24}{2} - 12$

Factor $2$ out of $24$ .

$f (2 x + 24) = \frac{2 (x) + 2 \cdot 12}{2} - 12$

Factor $2$ out of $2 (x) + 2 (12)$ .

$f (2 x + 24) = \frac{2 (x + 12)}{2} - 12$

Cancel the common factors.

Factor $2$ out of $2$ .

$f (2 x + 24) = \frac{2 (x + 12)}{2 (1)} - 12$

Cancel the common factor.

$f (2 x + 24) = \frac{2 (x + 12)}{2 \cdot 1} - 12$

Rewrite the expression.

$f (2 x + 24) = \frac{x + 12}{1} - 12$

Divide $x + 12$ by $1$ .

$f (2 x + 24) = x + 12 - 12$

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

Subtract $12$ from $12$ .

$f (2 x + 24) = x + 0$

Add $x$ and $0$ .

$f (2 x + 24) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 2 x + 24$ is the inverse of $f (x) = \frac{x}{2} - 12$ .

$f^{- 1} (x) = 2 x + 24$

Find the Inverse x/2-12

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