To determine the equation that describes the function in the table, we begin by analyzing the provided values:
- When \( x = 0, y = 3 \)
- When \( x = 1, y = 3.5 \)
- When \( x = 2, y = 4 \)
- When \( x = 3, y = 4.5 \)
The change in \( y \) corresponds to the change in \( x \). Observing the table:
- From \( x=0 \) to \( x=1 \): \( y \) changes from 3 to 3.5 (increase of 0.5)
- From \( x=1 \) to \( x=2 \): \( y \) changes from 3.5 to 4 (increase of 0.5)
- From \( x=2 \) to \( x=3 \): \( y \) changes from 4 to 4.5 (increase of 0.5)
This indicates that for every increase of 1 in \( x \), \( y \) increases by 0.5, suggesting a linear relationship. This can be expressed as:
\[ y = 0.5x + b \]
To find \( b \), we can use one of the points. Let's use the point \( (0, 3) \):
\[ 3 = 0.5(0) + b \implies b = 3 \]
Thus, the equation becomes:
\[ y = 0.5x + 3 \]
In function notation:
\[ f(x) = \frac{1}{2}x + 3 \]
Therefore, the correct function from the options provided is:
f(x) = \frac{1}{2} x + 3