To determine the [tex]$y$[/tex]-intercept of the function [tex]\( f(x) = x^2 + 3x + 5 \)[/tex], we need to find the value of the function when [tex]\( x = 0 \)[/tex].
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].
Substitute [tex]\( x = 0 \)[/tex] into the function [tex]\( f(x) = x^2 + 3x + 5 \)[/tex]:
[tex]\[
f(0) = (0)^2 + 3(0) + 5
\][/tex]
Simplify the expression:
[tex]\[
f(0) = 0 + 0 + 5 = 5
\][/tex]
Therefore, the [tex]$y$[/tex]-intercept of the function is the point where [tex]\( x = 0 \)[/tex] and [tex]\( y = 5 \)[/tex]. Hence, the [tex]$y$[/tex]-intercept is [tex]\((0, 5)\)[/tex].
So, the correct answer is:
[tex]\[
(0, 5)
\][/tex]
