X y

To solve this given the points, it appears you are trying to determine a function based on the input-output relationship you've provided. The points you presented are:

X  Y
0  3
1  3.5
2  4
3  4.5

From these points, we can see that as \( x \) increases by 1, \( y \) increases by 0.5.

This suggests a linear relationship.

Given the options provided:

1. \( f(x) = 3x + \frac{1}{2} \)
2. \( f(y) = 3x + \frac{1}{2} \)
3. \( f(x) = \frac{1}{2}x + 3 \)
4. \( f(y) = \frac{1}{2}x + 3 \)

Let's rewrite the output values for \( y \) in terms of \( x \):

The formula \( f(x) = \frac{1}{2}x + 3 \) can be tested:

• For \( x = 0 \): \( f(0) = \frac{1}{2}(0) + 3 = 3 \)
• For \( x = 1 \): \( f(1) = \frac{1}{2}(1) + 3 = 3.5 \)
• For \( x = 2 \): \( f(2) = \frac{1}{2}(2) + 3 = 4 \)
• For \( x = 3 \): \( f(3) = \frac{1}{2}(3) + 3 = 4.5 \)

Thus, this function matches all the provided points.

Based on this analysis, the answer is:

\( f(x) = \frac{1}{2}x + 3 \).