To determine the [tex]\( y \)[/tex]-intercept of the function [tex]\( g(x) \)[/tex], we need to evaluate [tex]\( g(x) \)[/tex] at [tex]\( x = 0 \)[/tex]. The [tex]\( y \)[/tex]-intercept is where the graph of the function crosses the [tex]\( y \)[/tex]-axis, which occurs when [tex]\( x = 0 \)[/tex].
Given the function [tex]\( g(x) = 2f(x) + 1 \)[/tex]:
1. To find the [tex]\( y \)[/tex]-intercept, we substitute [tex]\( x = 0 \)[/tex] into the function:
[tex]\[
g(0) = 2f(0) + 1
\][/tex]
2. Here, [tex]\( f(0) \)[/tex] represents the value of the function [tex]\( f(x) \)[/tex] at [tex]\( x = 0 \)[/tex]. This is a known value.
3. Next, substitute the assumed value [tex]\( f(0) = 1 \)[/tex]:
[tex]\[
g(0) = 2 \cdot 1 + 1 = 2 + 1 = 3
\][/tex]
Therefore, the [tex]\( y \)[/tex]-intercept of the function [tex]\( g(x) \)[/tex] is [tex]\((0, 3)\)[/tex].
The correct answer is:
C. [tex]\((0, 3)\)[/tex]
