Answer :
To determine the [tex]\( y \)[/tex]-intercept of the function [tex]\( g(x) = 2 f(x) + 1 \)[/tex] where [tex]\( f(x) = e^x \)[/tex], we need to evaluate [tex]\( g(x) \)[/tex] at [tex]\( x = 0 \)[/tex].
1. Substitute [tex]\( x = 0 \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(0) = e^0 \][/tex]
Since [tex]\( e^0 = 1 \)[/tex], we have:
[tex]\[ f(0) = 1 \][/tex]
2. Use the result from step 1 to evaluate [tex]\( g(x) \)[/tex] at [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = 2 f(0) + 1 \][/tex]
Substitute [tex]\( f(0) = 1 \)[/tex]:
[tex]\[ g(0) = 2 \cdot 1 + 1 = 2 + 1 = 3 \][/tex]
Thus, the [tex]\( y \)[/tex]-intercept of the function [tex]\( g(x) \)[/tex] is [tex]\((0, 3)\)[/tex].
Therefore, the correct answer is:
D. [tex]\((0, 3)\)[/tex]
1. Substitute [tex]\( x = 0 \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(0) = e^0 \][/tex]
Since [tex]\( e^0 = 1 \)[/tex], we have:
[tex]\[ f(0) = 1 \][/tex]
2. Use the result from step 1 to evaluate [tex]\( g(x) \)[/tex] at [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = 2 f(0) + 1 \][/tex]
Substitute [tex]\( f(0) = 1 \)[/tex]:
[tex]\[ g(0) = 2 \cdot 1 + 1 = 2 + 1 = 3 \][/tex]
Thus, the [tex]\( y \)[/tex]-intercept of the function [tex]\( g(x) \)[/tex] is [tex]\((0, 3)\)[/tex].
Therefore, the correct answer is:
D. [tex]\((0, 3)\)[/tex]
