Determine if a function is exponential, linear, or neither by analyzing the pattern of f(x) values in tables.

Let's analyze each table (a), (b), and (c) to determine whether the function is exponential, linear, or neither. We'll look at the pattern of change in the output values $ f(x) $ as $ x $ increases by 1.

---

a)

| x | f(x) |
|---|------|
| 1 | 12 |
| 2 | 17 |
| 3 | 22 |
| 4 | 27 |
| 5 | 32 |

Step 1: Check differences between consecutive f(x) values

- $ 17 - 12 = 5 $
- $ 22 - 17 = 5 $
- $ 27 - 22 = 5 $
- $ 32 - 27 = 5 $

✔ The first differences are constant (always +5).

This indicates a linear function.

> ✔ Answer: Linear
> Reason: Constant rate of change (+5 each time).
> This matches what’s already written: *"Linear, adding by 5 each time."*

---

b)

| x | f(x) |
|---|------|
| 1 | 2 |
| 2 | 7 |
| 3 | 13 |
| 4 | 20 |
| 5 | 28 |

Step 1: First differences

- $ 7 - 2 = 5 $
- $ 13 - 7 = 6 $
- $ 20 - 13 = 7 $
- $ 28 - 20 = 8 $

First differences: 5, 6, 7, 8 → not constant → not linear

Step 2: Second differences (differences of first differences)

- $ 6 - 5 = 1 $
- $ 7 - 6 = 1 $
- $ 8 - 7 = 1 $

Second differences are constant (all 1). This suggests a quadratic relationship — but we're only asked for exponential, linear, or neither.

Since it's not linear and doesn't have a constant ratio (which would suggest exponential), let's check ratios:

Check ratios (for exponential):

- $ 7 / 2 = 3.5 $
- $ 13 / 7 ≈ 1.857 $
- $ 20 / 13 ≈ 1.538 $
- $ 28 / 20 = 1.4 $

Ratios are not constant → not exponential

So:
- Not linear (first differences not constant)
- Not exponential (ratios not constant)

> ✔ Answer: Neither

---

c)

| x | f(x) |
|---|------|
| 1 | 1/3 |
| 2 | 1 |
| 3 | 3 |
| 4 | 9 |
| 5 | 27 |

Step 1: Look at ratios (since values are growing fast)

- $ 1 / (1/3) = 3 $
- $ 3 / 1 = 3 $
- $ 9 / 3 = 3 $
- $ 27 / 9 = 3 $

✔ Ratios are constant: ×3 each time

This is a hallmark of an exponential function.

We can write this as:
$$
f(x) = \left(\frac{1}{3}\right) \cdot 3^{x-1}
\quad \text{or} \quad f(x) = 3^{x-2}
$$
But regardless, the pattern is exponential.

> ✔ Answer: Exponential

---

✔ Final Answers:

- a) Linear
- b) Neither
- c) Exponential

Let me know if you'd like to see the equations!