To find the possible values of \( k \) for the given function table, we can examine the relationships between the values of \( x \) and \( y \). The input values (x) and corresponding output values (y) are given as follows:
- \( (4, 0) \)
- \( (-2, 1) \)
- \( (1, 6) \)
- \( (3, -1) \)
- \( (-5, 2) \)
- \( (k, 3) \)
Since we are looking for a potential function, let's first determine if there is a consistent pattern or a possible function that might relate \( x \) to \( y \).
We can observe the differences or try to derive a linear or polynomial function. One approach is to check the possibility of a linear relationship of the form \( y = mx + b \).
Let's take two points and calculate the slope \( m \):
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From points \( (-2, 1) \) and \( (4, 0) \): \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 1}{4 - (-2)} = \frac{-1}{6} = -\frac{1}{6} \] Now, we can use one of these points to solve for \( b \): \[ 1 = -\frac{1}{6}(-2) + b \Rightarrow 1 = \frac{1}{3} + b \Rightarrow b = 1 - \frac{1}{3} = \frac{2}{3} \]
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Therefore, the potential equation can be: \[ y = -\frac{1}{6}x + \frac{2}{3} \]
Now, we would use this equation to check if it can allow for \( y = 3 \): \[ 3 = -\frac{1}{6}k + \frac{2}{3} \] To isolate \( k \): \[ 3 - \frac{2}{3} = -\frac{1}{6}k \Rightarrow \frac{9}{3} - \frac{2}{3} = -\frac{1}{6}k \Rightarrow \frac{7}{3} = -\frac{1}{6}k \] Multiplying both sides by -6 to solve for \( k \): \[ k = -6 \cdot \frac{7}{3} = -14 \]
Thus, a possible value for \( k \) is \( -14 \). However, to confirm \( k \), we should verify if there are other points in the table that lead to the same. Reviewing other points with their respective values of \( y \), if no contradictions arise, then indeed \( k = -14 \) is a valid rational solution.
Hence, the possible value of \( k \) is: \[ \boxed{-14} \]
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