Question

24.
Write a function rule for the table.

x ƒ(x)
3 –1
4 0
5 1
6 2

Answer 1

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:**

\[
f(x) = x - 4
\]