## Explanation

To find the inverse of the given equation:

$(x - 4)^2 - \frac{2}{3} = 6y - 12$### Step 1: Simplify the Original Equation

First, we start by isolating $y$ on one side of the equation. Add $12$ to both sides:

$(x - 4)^2 - \frac{2}{3} + 12 = 6y$Simplify:

$(x - 4)^2 + 11\frac{1}{3} = 6y$Note that:

$11\frac{1}{3} = \frac{34}{3}$Thus:

$(x - 4)^2 + \frac{34}{3} = 6y$Multiply through by 3:

$3(x - 4)^2 + 34 = 18y$Divide both sides by 18:

$y = \frac{1}{18}[3(x - 4)^2 + 34]$### Step 2: Solve for $x$ in Terms of $y$

Swap $x$ and $y$:

$x = \frac{1}{18}[3(y - 4)^2 + 34]$Multiply by 18:

$18x = 3(y - 4)^2 + 34$Subtract 34 from both sides:

$18x - 34 = 3(y - 4)^2$Divide by 3:

$6x - \frac{34}{3} = (y - 4)^2$### Step 3: Rewrite the Inverse Equation

Finally, solve for $y$:

$(y - 4)^2 = 6x - \frac{34}{3}$Take the square root of both sides:

$y - 4 = \pm \sqrt{6x - \frac{34}{3}}$Add 4 to both sides:

$y = 4 \pm \sqrt{6x - \frac{34}{3}}$**Thus, the inverse function is:**

This gives us both possible values for the inverse function, depending on whether the positive or negative square root is taken.