https://file.aiquickdraw.com/mathbot/file/1734129216928ssy3u3zc.jpg

Tell whether each function is linear or exponential.

To determine whether the function is linear or exponential, let's analyze the data in the table:

### Table:
| \( x \) | \( f(x) \) |
|--------|-----------|
| 0      | 5         |
| 1      | 9         |
| 2      | 13        |
| 3      | 17        |
| 4      | 21        |

#### Step 1: Check for a constant difference (linear function)
- \( f(1) - f(0) = 9 - 5 = 4 \)
- \( f(2) - f(1) = 13 - 9 = 4 \)
- \( f(3) - f(2) = 17 - 13 = 4 \)
- \( f(4) - f(3) = 21 - 17 = 4 \)

Since the differences between successive outputs (\( f(x) \)) are constant, the function is **linear**.

#### Step 2: Confirm it is not exponential
For exponential functions, the ratio between consecutive outputs must be constant:
- \( \frac{f(1)}{f(0)} = \frac{9}{5} = 1.8 \)
- \( \frac{f(2)}{f(1)} = \frac{13}{9} \neq 1.8 \)

The ratio is not constant, so the function is not exponential.

### Conclusion:
The function is **linear**.

Would you like a detailed explanation or a formula for the function? Let me know! 

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#### Related Questions:
1. How can we derive the equation of this linear function?
2. What are the differences between linear and exponential functions?
3. What would the graph of this function look like?
4. How do you test whether a function is quadratic instead of linear?
5. Can you create a new table where the function is exponential?

#### Tip:
To identify a linear function, always start by checking the differences between outputs—they must be constant!

Learn how to determine whether a function is linear or exponential using a table of values. This example shows a function where differences are constant, confirming it is linear. Perfect for Grades 7-9 students learning function types.

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