To determine the [tex]\( y \)[/tex]-intercept of the function [tex]\( g(x) = x^2 + 2 \)[/tex], we follow these steps:
1. Identify the point of interest: The [tex]\( y \)[/tex]-intercept occurs where the graph of the function crosses the [tex]\( y \)[/tex]-axis. This happens when [tex]\( x = 0 \)[/tex].
2. Substitute [tex]\( x = 0 \)[/tex] into the function: To find the [tex]\( y \)[/tex]-intercept, we evaluate the function [tex]\( g(x) \)[/tex] at [tex]\( x = 0 \)[/tex].
3. Calculate [tex]\( g(0) \)[/tex]:
[tex]\[
g(0) = 0^2 + 2
\][/tex]
[tex]\[
g(0) = 0 + 2
\][/tex]
[tex]\[
g(0) = 2
\][/tex]
4. Determine the coordinates of the [tex]\( y \)[/tex]-intercept: When [tex]\( x = 0 \)[/tex], [tex]\( g(x) = 2 \)[/tex]. Thus, the [tex]\( y \)[/tex]-intercept is at the point [tex]\((0, 2)\)[/tex].
Based on this, the correct answer is:
[tex]\[ \boxed{(0,2)} \][/tex]
