Question

Compare the graphs of f(x)=2√x+1 and g(x)=2^3√x+1
. What is the y-intercept of both graphs?
Responses

(2, 0)

(-1, 0)

(0, 2)

(0, -1)

Answer 1

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 \( f(x) = 2\sqrt{x} + 1 \):

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 \( g(x) = 2^{3\sqrt{x} + 1} \):

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.)