To find the y-intercepts of the functions \( f(x) = 2\sqrt{x} + 1 \) and \( g(x) = 2^{3\sqrt{x} + 1} \), we need to evaluate each function at \( x = 0 \).
For \( f(x) = 2\sqrt{x} + 1 \):
- Plug \( x = 0 \) into the function: \[ f(0) = 2\sqrt{0} + 1 = 2(0) + 1 = 1 \] So, the y-intercept of \( f(x) \) is \( (0, 1) \).
For \( g(x) = 2^{3\sqrt{x} + 1} \):
- Plug \( x = 0 \) into the function: \[ g(0) = 2^{3\sqrt{0} + 1} = 2^{3(0) + 1} = 2^1 = 2 \] So, the y-intercept of \( g(x) \) is \( (0, 2) \).
Summary of y-intercepts:
- The y-intercept of \( f(x) \) is \( (0, 1) \).
- The y-intercept of \( g(x) \) is \( (0, 2) \).
From the available response options, the closest match to the y-intercept of \( g(x) \) is:
- \( (0, 2) \)
Thus, the final answer regarding the y-intercepts of both graphs is:
- For \( g(x) \): (0, 2)
(Note: \( (0, 1) \) is not one of the provided options.)