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 \( y \) in the original equation \( y = 2x^2 - 8 \). So 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, so \( y=\pm\sqrt{\frac{x + 8}{2}} \).

Answer:

\( y = \pm\sqrt{\frac{x + 8}{2}} \) (the first option: \( y=\pm\sqrt{\frac{x + 8}{2}} \))