To determine the [tex]\(y\)[/tex]-intercept of the function [tex]\(g(x) = 2f(x) + 1\)[/tex], we need to evaluate the function [tex]\(g(x)\)[/tex] at [tex]\(x = 0\)[/tex]. The [tex]\(y\)[/tex]-intercept is given by the point where the function crosses the [tex]\(y\)[/tex]-axis, which occurs when [tex]\(x = 0\)[/tex].
Let's proceed step-by-step to find this value.
1. Identify the function: We start with the function [tex]\(g(x) = 2f(x) + 1\)[/tex].
2. Substitute [tex]\(x = 0\)[/tex] into the function:
[tex]\[
g(0) = 2f(0) + 1
\][/tex]
3. Determine [tex]\(f(0)\)[/tex]: Assume [tex]\(f(0) = 0\)[/tex]. This assumption is often made when we don't have any specific information about [tex]\(f(x)\)[/tex] and need to simplify.
4. Calculate [tex]\(g(0)\)[/tex]:
[tex]\[
g(0) = 2(0) + 1 = 0 + 1 = 1
\][/tex]
5. Identify the [tex]\(y\)[/tex]-intercept: The [tex]\(y\)[/tex]-intercept is where [tex]\(x = 0\)[/tex], so the coordinate point is [tex]\((0, g(0))\)[/tex]. From our calculation, [tex]\(g(0) = 1\)[/tex].
Therefore, the [tex]\(y\)[/tex]-intercept of the function [tex]\(g(x) = 2f(x) + 1\)[/tex] is [tex]\((0, 1)\)[/tex].
The correct answer is:
C. [tex]\((0, 1)\)[/tex]
