Question

9.
Write a function rule for the table.

x ƒ(x)
3 –1
4 0
5 1
6 2

Answer 1

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:

\[
\boxed{f(x) = x - 4}
\]