Linear function A and linear function B both have the same input values as shown below. Why will the output values for linear function A always be different than the corresponding output values for linear function B?

To understand why the output values for linear function A will always be different from the corresponding output values for linear function B, we can analyze the two functions based on the provided tables.

Let's summarize the information given:

- **Linear Function A:**
- Input (x): 1, 3, 5, 7, 9
- Output (y): 3, 7, 11, 15, 19

- **Linear Function B:**
- Input (x): 1, 3, 5, 7, 9
- Output (y): 4, 8, 12, 16, 20

Now, we can clearly see from the tables:

1. **Initial Values (y-intercepts):**
- For Linear Function A, when \(x = 1\), \(y = 3\).
- For Linear Function B, when \(x = 1\), \(y = 4\).
- The initial values (or outputs for the first input) are different (3 for A and 4 for B).

2. **Rates of Change (slopes):**
- The change in output for Linear Function A (between successive input values) is consistently 4 (e.g., \(7 - 3 = 4\), \(11 - 7 = 4\), etc.),
- The change in output for Linear Function B is also consistently 4 (e.g., \(8 - 4 = 4\), \(12 - 8 = 4\), etc.).
- Therefore, both functions have the same rate of change (slope is the same).

Given these observations:

- The initial values (or outputs) for the two functions are different.
- However, the rates of change are the same.

Thus, the appropriate conclusion is:

**The initial values of the two functions are different, and the rates of change of the two functions are also the same.**

This is the reason why the output values for linear function A will always be different from the corresponding output values for linear function B.