Question

Is this function linear or exponential?
x y
0
–
2
1
–
4
2
–
8
3
–
16
4
–
3

Answer 1

To determine whether the function described by the data is linear or exponential, we can analyze the relationship between the x and y values.

From the data provided:

```
x y
0 –
1 4
2 8
3 16
4 3
```

However, the value for \( y \) when \( x = 0 \) is not specified and appears to be incomplete or marked with a dash. For analysis, let's focus on the other given values.

### Analyzing the y-values:

1. **At x = 1, y = 4**
2. **At x = 2, y = 8**
3. **At x = 3, y = 16**
4. **At x = 4, y = 3**

### Checking for a linear relationship:
A linear function has a constant rate of change (additive difference) between successive y-values.

Calculating the differences between the y-values:

- \( y(2) - y(1) = 8 - 4 = 4 \)
- \( y(3) - y(2) = 16 - 8 = 8 \)
- \( y(4) - y(3) = 3 - 16 = -13 \)

The differences are not constant; they change from 4 to 8 to -13. Thus, this relationship is not linear.

### Checking for an exponential relationship:
An exponential function has a constant ratio between successive y-values (multiplicative change).

Calculating the ratios:

- \( \frac{y(2)}{y(1)} = \frac{8}{4} = 2 \)
- \( \frac{y(3)}{y(2)} = \frac{16}{8} = 2 \)
- \( \frac{y(4)}{y(3)} = \frac{3}{16} = 0.1875 \)

The ratios are not constant; they are 2, 2, and approximately 0.1875, indicating that the function is not exponential.

### Conclusion:
Since the function does not exhibit a constant difference (not linear) and does not exhibit a constant ratio (not exponential), this dataset does not fit either a linear or exponential model based solely on the provided values. Further context on the behavior of the function or additional points could help clarify its nature.