Find the Inverse (x-6)/2

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

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

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

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

Multiply both sides of the equation by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$y - 6 = 2 x$

Add $6$ to both sides of the equation.

$y = 2 x + 6$

Step 3

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

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

Step 4

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

Add $- 6$ and $6$ .

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

Add $x$ and $0$ .

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

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

Set up the composite result function.

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

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

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

Cancel the common factor of $(2 x + 6) - 6$ and $2$ .

Factor $2$ out of $2 x$ .

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

Factor $2$ out of $6$ .

$f (2 x + 6) = \frac{2 (x) + 2 \cdot 3 - 6}{2}$

Factor $2$ out of $2 (x) + 2 (3)$ .

$f (2 x + 6) = \frac{2 (x + 3) - 6}{2}$

Factor $2$ out of $- 6$ .

$f (2 x + 6) = \frac{2 (x + 3) + 2 \cdot - 3}{2}$

Factor $2$ out of $2 (x + 3) + 2 (- 3)$ .

$f (2 x + 6) = \frac{2 (x + 3 - 3)}{2}$

Cancel the common factors.

Factor $2$ out of $2$ .

$f (2 x + 6) = \frac{2 (x + 3 - 3)}{2 (1)}$

Cancel the common factor.

$f (2 x + 6) = \frac{2 (x + 3 - 3)}{2 \cdot 1}$

Rewrite the expression.

$f (2 x + 6) = \frac{x + 3 - 3}{1}$

Divide $x + 3 - 3$ by $1$ .

$f (2 x + 6) = x + 3 - 3$

Combine the opposite terms in $x + 3 - 3$ .

Subtract $3$ from $3$ .

$f (2 x + 6) = x + 0$

Add $x$ and $0$ .

$f (2 x + 6) = x$

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

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

Find the Inverse (x-6)/2

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