To find the [tex]\( y \)[/tex]-intercept of the function [tex]\( g(x) = 2f(x) + 1 \)[/tex], where [tex]\( f(x) = e^x \)[/tex], we need to determine the value of [tex]\( g(x) \)[/tex] when [tex]\( x = 0 \)[/tex].
First, substitute [tex]\( x = 0 \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(0) = e^0 \][/tex]
Since [tex]\( e^0 = 1 \)[/tex]:
[tex]\[ f(0) = 1 \][/tex]
Next, substitute [tex]\( f(0) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g(0) = 2f(0) + 1 \][/tex]
[tex]\[ g(0) = 2 \cdot 1 + 1 \][/tex]
[tex]\[ g(0) = 2 + 1 \][/tex]
[tex]\[ g(0) = 3 \][/tex]
Therefore, the [tex]\( y \)[/tex]-intercept of the function [tex]\( g(x) \)[/tex] is [tex]\( (0, 3) \)[/tex].
The correct answer is:
B. [tex]\((0, 3)\)[/tex]
