To find the function rule for the given table, we can analyze the relationship between the input values \( x \) and the corresponding output values \( f(x) \).
Here's the provided data:
| x | f(x) | |---|------| | 3 | -1 | | 4 | 0 | | 5 | 1 | | 6 | 2 |
We can observe the following changes in \( f(x) \):
- 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, \( f(x) \) also increases from 0 to 1 (an increase of 1).
- When \( x \) increases from 5 to 6, \( f(x) \) again increases from 1 to 2 (an increase of 1).
From this observation, we can see that for each increment of 1 in \( x \), \( f(x) \) increases by 1. This suggests that \( f(x) \) is a linear function with a slope of 1.
Next, let's derive the function rule. We can express \( f(x) \) in the form:
\[ f(x) = mx + b \]
where \( m \) is the slope and \( b \) is the y-intercept.
We already established that \( m = 1 \).
Now, let's find the value of \( b \). We can use one of the points from the table. Let's use the point \( (4, 0) \):
\[ 0 = 1(4) + b \] \[ 0 = 4 + b \] \[ b = -4 \]
Now that we have \( m = 1 \) and \( b = -4 \), we can write the function rule:
\[ f(x) = x - 4 \]
We can verify this function by substituting the values of \( x \) from the table:
- \( f(3) = 3 - 4 = -1 \)
- \( f(4) = 4 - 4 = 0 \)
- \( f(5) = 5 - 4 = 1 \)
- \( f(6) = 6 - 4 = 2 \)
All values match the original table, confirming that the function rule is correct.
Final function rule:
\[ f(x) = x - 4 \]