To find the inverse of the equation [tex]\( y = x^2 - 36 \)[/tex], follow these steps:
1. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]: The inverse function means we want to solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]. To do this, we first swap [tex]\( y \)[/tex] and [tex]\( x \)[/tex]:
[tex]\[
x = y^2 - 36
\][/tex]
2. Solve for [tex]\( y \)[/tex]: To solve for [tex]\( y \)[/tex], isolate [tex]\( y \)[/tex] on one side of the equation:
[tex]\[
x = y^2 - 36
\][/tex]
Add 36 to both sides:
[tex]\[
x + 36 = y^2
\][/tex]
3. Take the square root: To solve for [tex]\( y \)[/tex], take the square root of both sides. Remember to include both the positive and negative roots because [tex]\( y \)[/tex] could be either positive or negative:
[tex]\[
y = \pm \sqrt{x + 36}
\][/tex]
So the inverse function of [tex]\( y = x^2 - 36 \)[/tex] is:
[tex]\[
y = \pm \sqrt{x + 36}
\][/tex]
Among the provided multiple choices, the correct option is:
[tex]\[
\boxed{2. \, y = \pm \sqrt{x + 36}}
\][/tex]
