To find the inverse of the equation
\((x-4)^2 - \frac{2}{3} - 6y - 12\),
we need to rearrange it so that \( y \) is expressed in terms of \( x \). Let's go through the steps:
Step 1: Start with the original equation:
[tex]\[
(x-4)^2 - \frac{2}{3} - 6y - 12 = 0
\][/tex]
Step 2: Isolate the \( y \)-related terms on one side of the equation:
[tex]\[
(x-4)^2 - \frac{2}{3} - 12 = 6y
\][/tex]
Step 3: Combine constants on the left side:
[tex]\[
(x-4)^2 - \frac{2}{3} - 12 = (x-4)^2 - 12.6667 (or \frac{38}{3})
\][/tex]
Step 4: Combine the constant terms:
[tex]\[
(x-4)^2 - \frac{40}{3} = 6y
\][/tex]
Step 5: Divide both sides of the equation by 6 to solve for \( y \):
[tex]\[
y = \frac{1}{6} (x-4)^2 - \frac{40}{6 \cdot 3}
\][/tex]
Given the options, the correct and simplified inverse is:
[tex]\[
y=-4 \pm \sqrt{6 x-\frac{34}{3}}
\][/tex]
Therefore, the equation [tex]\( y=-4 \pm \sqrt{6 x-\frac{34}{3}} \)[/tex] is the inverse of the given equation.
