To find the [tex]\( y \)[/tex]-intercept of the function [tex]\( g \)[/tex] where [tex]\( g(x) = 2f(x) + 1 \)[/tex] and [tex]\( f(x) = e^x \)[/tex], we need to determine the value of [tex]\( g(x) \)[/tex] when [tex]\( x = 0 \)[/tex].
Step-by-step solution:
1. Identify the given functions:
[tex]\[
f(x) = e^x
\][/tex]
[tex]\[
g(x) = 2f(x) + 1
\][/tex]
2. Evaluate [tex]\( f(x) \)[/tex] at [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = e^0 = 1
\][/tex]
3. Plug [tex]\( f(0) \)[/tex] into the function [tex]\( g(x) \)[/tex]:
[tex]\[
g(0) = 2f(0) + 1 = 2 \cdot 1 + 1 = 2 + 1 = 3
\][/tex]
4. Conclusion:
The [tex]\( y \)[/tex]-intercept of the function [tex]\( g(x) \)[/tex] occurs at the point where [tex]\( x = 0 \)[/tex]. So, the [tex]\( y \)[/tex]-intercept is:
[tex]\[
(0, 3)
\][/tex]
Therefore, the correct answer is:
A. [tex]\((0, 3)\)[/tex]
