To determine which equation describes the function in the table, we first need to understand the relationship between \(x\) and \(y\) based on the values given:
- For \(x = 0\), \(y = 3\)
- For \(x = 1\), \(y = 312\)
- For \(x = 2\), \(y = 4\)
- For \(x = 3\), \(y = 412\)
There seems to be some inconsistencies in the table which could be a typographical error (like the value 312 for \(y\)). However, if we focus on the values we have, it appears that the correct pairs are likely:
- \( (0, 3) \)
- \( (2, 4) \)
Assuming a linear relationship, let's try to determine the linear function based on the correct pairs:
Using points \( (0, 3) \) and \( (2, 4) \):
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Calculate the slope \(m\): \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 3}{2 - 0} = \frac{1}{2} \]
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Using point-slope form, where \(y - y_1 = m(x - x_1)\) and using the point \( (0, 3) \): \[ y - 3 = \frac{1}{2}(x - 0) \] \[ y = \frac{1}{2}x + 3 \]
In function notation, this can be written as: \[ f(x) = \frac{1}{2}x + 3 \]
Thus, based on the options provided, the correct response is: f(x)=Start Fraction 1 over 2 End Fraction x + 3.