Question

which equation is the inverse of ((x - 4)^2 - \frac{2}{3} = 6y - 12)?
(y = \frac{1}{6}x^2 - \frac{4}{3}x + \frac{43}{9})
(y = 4 pm sqrt{6x - \frac{34}{3}})
(-(x - 4)^2 - \frac{2}{3} = -6y + 12)

Explanation:

Step1: Swap x and y

To find the inverse of a function, we first swap \( x \) and \( y \) in the original equation. The original equation is \( (x - 4)^2 - \frac{2}{3} = 6y - 12 \). After swapping, we get:
\[
(y - 4)^2 - \frac{2}{3} = 6x - 12
\]

Step2: Solve for y

First, add \( \frac{2}{3} \) to both sides:
\[
(y - 4)^2 = 6x - 12 + \frac{2}{3}
\]
Simplify the right - hand side: \( - 12+\frac{2}{3}=\frac{-36 + 2}{3}=\frac{-34}{3}\), so \( (y - 4)^2=6x-\frac{34}{3} \)
Then, take the square root of both sides:
\[
y - 4=\pm\sqrt{6x-\frac{34}{3}}
\]
Finally, add 4 to both sides:
\[
y = 4\pm\sqrt{6x-\frac{34}{3}}
\]

Answer:

\( y = 4\pm\sqrt{6x-\frac{34}{3}} \) (the second option in the given choices)