Question

How to find the solution to a function given input-output pairs (4,9), (6,11), and (9,14)?

Answer 1

To find a function that fits the given input-output pairs \((4,9)\), \((6,11)\), and \((9,14)\), we should analyze the relationship between the inputs and outputs. Let's denote the input as \(x\) and the output as \(y\).

The pairs are:
1. \( (4, 9) \)
2. \( (6, 11) \)
3. \( (9, 14) \)

1. **Identify the differences in outputs**:
- For \(x = 4\) to \(x = 6\):
\[
11 - 9 = 2
\]
- For \(x = 6\) to \(x = 9\):
\[
14 - 11 = 3
\]

2. **Identify differences in inputs**:
- From \(x = 4\) to \(x = 6\) (a change of 2), the output increases by 2.
- From \(x = 6\) to \(x = 9\) (a change of 3), the output increases by 3.

Based on the output differences, we may suspect that the function could be linear. Let's check and see if a linear function of the form \(y = mx + b\) fits.

3. **Calculate the slope \(m\)**:
The slope between the first two points \((4, 9)\) and \((6, 11)\):
\[
m = \frac{11 - 9}{6 - 4} = \frac{2}{2} = 1
\]

4. **Use slope \(m\) to get the equation**:
Let’s assume the form \(y = x + b\).
Substituting one point, say \((4, 9)\):
\[
9 = 4 + b \implies b = 9 - 4 = 5
\]
So we have the tentative equation:
\[
y = x + 5
\]

5. **Check the other points**:
- For \(x = 6\):
\[
y = 6 + 5 = 11 \quad \text{(Matches)}
\]
- For \(x = 9\):
\[
y = 9 + 5 = 14 \quad \text{(Matches)}
\]

All points satisfy the equation \(y = x + 5\).

**Conclusion:**
The function that fits the input-output pairs \((4,9)\), \((6,11)\), and \((9,14)\) is:
\[
y = x + 5
\]