To create an exponential model from the dataset provided, we can observe the pattern in the y-values as the x-values increase.
The x-values and corresponding y-values given are:
- For \( x = 0 \), \( y = 6 \)
- For \( x = 1 \), \( y = 12 \)
- For \( x = 2 \), \( y = 24 \)
- For \( x = 3 \), \( y = 48 \)
Notice that:
- From \( y = 6 \) to \( y = 12 \): \( 12 = 6 \times 2 \)
- From \( y = 12 \) to \( y = 24 \): \( 24 = 12 \times 2 \)
- From \( y = 24 \) to \( y = 48 \): \( 48 = 24 \times 2 \)
This reveals that each y-value is multiplied by 2 as x increases by 1. Therefore, it appears that the function can be of the form \( f(x) = a \cdot 2^x \).
Given that when \( x = 0 \), \( f(0) = 6 \), we can deduce:
\[ f(0) = a \cdot 2^0 = a = 6 \]
Thus the exponential model can be written as:
\[ f(x) = 6 \cdot 2^x \]
Therefore, the correct answer is:
f(x) = 6(2^x)