To find the inverse of the function [tex]\( y = x^2 + 16 \)[/tex], we need to express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex].
1. Start with the given function:
[tex]\[
y = x^2 + 16
\][/tex]
2. To find the inverse, switch the roles of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
x = y^2 + 16
\][/tex]
3. Solve this equation for [tex]\( y \)[/tex]:
[tex]\[
x - 16 = y^2
\][/tex]
4. Isolate [tex]\( y \)[/tex] by taking the square root of both sides. Remember, taking the square root of both sides introduces a [tex]\(\pm\)[/tex] (plus-minus) sign:
[tex]\[
y = \pm \sqrt{x - 16}
\][/tex]
Thus, the inverse function is:
[tex]\[
y = \pm \sqrt{x - 16}
\][/tex]
Therefore, the correct option corresponding to the inverse function [tex]\( y = \pm \sqrt{x - 16} \)[/tex] is the third option:
[tex]\[
\boxed{y = \pm \sqrt{x - 16}}
\][/tex]
