To identify the correct equation that describes the function given in the table, we first need to examine the x and y values provided:
- When \( x = 0 \), \( y = 3 \)
- When \( x = 1 \), \( y = 3.5 \)
- When \( x = 2 \), \( y = 4 \)
- When \( x = 3 \), \( y = 4.5 \)
From this data, we can observe the following changes in \( y \) with respect to \( x \):
- From \( x = 0 \) to \( x = 1 \) (increase of 1): \( y \) increases from 3 to 3.5 (an increase of 0.5)
- From \( x = 1 \) to \( x = 2 \) (increase of 1): \( y \) increases from 3.5 to 4 (an increase of 0.5)
- From \( x = 2 \) to \( x = 3 \) (increase of 1): \( y \) increases from 4 to 4.5 (an increase of 0.5)
This indicates a linear relationship where \( y \) increases by 0.5 for each increase of 1 in \( x \).
To express this relationship in slope-intercept form (which is \( y = mx + b \)), we note:
- The slope \( m \) is 0.5 (since \( 4.5 - 3 = 1.5 \) over the change in \( x = 3 - 0 = 3 \), gives a slope of \( \frac{1.5}{3} = 0.5 \)).
- The y-intercept \( b \) is 3 (when \( x = 0 \)).
Thus, the equation can be derived as: \[ y = 0.5x + 3 \] or, converting the slope to a fraction, \[ y = \frac{1}{2} x + 3 \]
Using function notation, this can be written as: \[ f(x) = \frac{1}{2}x + 3 \]
However, simplified further, it can also be expressed as: \[ f(x) = \frac{1}{2} x + 3 \]
Now inspecting the given responses, we confirm:
- f(x)=3x+1/2 (incorrect)
- f(y)=12x+3 (incorrect)
- f(x)=12x+3 (incorrect)
- f(y)=3x+1/2 (incorrect)
Considering that there might be a typographical error in the options presented and what we derived matches none exactly, none of the responses appear correct based on the provided data.
If there should have been an option such as: \[ f(x) = \frac{1}{2} x + 3 \] or similar with correct formatting, it would have been correct. Make sure to recheck the function choices given or clarify if there was an input error.