To find a function rule that describes the relationship between \( x \) and \( f(x) \) based on the given table, we can look for a pattern in the output values.
We have the following pairs:
\[ \begin{align*} x = 3 & \implies f(x) = -1 \ x = 4 & \implies f(x) = 0 \ x = 5 & \implies f(x) = 1 \ x = 6 & \implies f(x) = 2 \ \end{align*} \]
From this data, we can observe how \( f(x) \) increases as \( x \) increases:
- When \( x \) increases from 3 to 4, \( f(x) \) increases from -1 to 0 (an increase of 1).
- When \( x \) increases from 4 to 5, \( f(x) \) increases from 0 to 1 (again an increase of 1).
- When \( x \) increases from 5 to 6, \( f(x) \) increases from 1 to 2 (once more, an increase of 1).
We see that \( f(x) \) increases by 1 for every increase of 1 in \( x \). This suggests that \( f(x) \) is a linear function.
To express this as a function rule, we can note the following:
- When \( x = 3 \), \( f(x) = -1 \).
- We can use the point \( (3, -1) \) to find the function rule.
Given the relationship we see, we can deduce that:
\[ f(x) = x - 4 \]
This works because:
- For \( x = 3 \): \( f(3) = 3 - 4 = -1 \)
- For \( x = 4 \): \( f(4) = 4 - 4 = 0 \)
- For \( x = 5 \): \( f(5) = 5 - 4 = 1 \)
- For \( x = 6 \): \( f(6) = 6 - 4 = 2 \)
Thus, the function rule that describes the relationship is:
\[ \boxed{f(x) = x - 4} \]