Which equation is the inverse of \((x-4)^2 - \frac{2}{3} - 6y - 12\)?

A. \(y = \frac{1}{6}x^2 - \frac{4}{3}x + \frac{43}{9}\)

B. \(y = 4 \pm \sqrt{6x - \frac{34}{3}}\)

C. \(y = -4 \pm \sqrt{6x - \frac{34}{3}}\)

D. [tex]\(-(x-4)^2 - \frac{2}{3} = -6y + 12\)[/tex]



Answer :

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.