Question

which equation is the inverse of $y = 2x^2 - 8$?
$\bigcirc\\ y = \pm\sqrt{\frac{x + 8}{2}}$
$\bigcirc\\ y = \frac{\pm\sqrt{x + 8}}{2}$
$\bigcirc\\ y = \pm\sqrt{\frac{x}{2} + 8}$
$\bigcirc\\ y = \frac{\pm\sqrt{x}}{2} + 4$

Explanation:

Step1: Swap x and y

To find the inverse of a function, we first swap the roles of \( x \) and \( y \). So for the equation \( y = 2x^2 - 8 \), we get \( x = 2y^2 - 8 \).

Step2: Solve for y

First, we isolate the term with \( y^2 \). Add 8 to both sides of the equation:
\( x + 8 = 2y^2 \)
Then, divide both sides by 2:
\( \frac{x + 8}{2} = y^2 \)
To solve for \( y \), we take the square root of both sides. Remember that when taking the square root, we need to consider both the positive and negative roots:
\( y = \pm \sqrt{\frac{x + 8}{2}} \)

Answer:

\( y = \pm \sqrt{\frac{x + 8}{2}} \) (which corresponds to the first option: \( y=\pm\sqrt{\frac{x + 8}{2}} \) or written as \( y=\pm\frac{\sqrt{x + 8}}{\sqrt{2}} \) but simplified as \( y=\pm\sqrt{\frac{x + 8}{2}} \), and among the given options, the first one is \( y=\pm\sqrt{\frac{x + 8}{2}} \) (the first option's format is \( y=\pm\sqrt{\frac{x + 8}{2}} \) when written properly with the fraction under the square root as \( \frac{x + 8}{2} \)) so the correct option is the first one: \( y=\pm\sqrt{\frac{x + 8}{2}} \) (the first option in the list: \( y=\pm\sqrt{\frac{x + 8}{2}} \))