To determine the function that best models the data and calculate the value of the function at \( x = -1 \), we can first examine the provided data:
| x-Values | y-Values | |----------|----------| | 0 | 5 | | 1 | 15 | | 2 | 45 | | 3 | 135 | | 4 | 405 |
From observing the values, we can see that:
- \( y(0) = 5 \)
- \( y(1) = 15 = 5 \cdot 3 \)
- \( y(2) = 45 = 5 \cdot 3^2 \)
- \( y(3) = 135 = 5 \cdot 3^3 \)
- \( y(4) = 405 = 5 \cdot 3^4 \)
This suggests that the function could be modeled by the equation \( f(x) = 5 \cdot 3^x \).
Now, let's confirm if this function fits and calculate \( f(-1) \):
\[ f(-1) = 5 \cdot 3^{-1} = 5 \cdot \frac{1}{3} = \frac{5}{3} \]
So the model \( f(x) = 5 \cdot 3^x \) is appropriate, and at \( x = -1 \), the function evaluates to:
\[ f(-1) = \frac{5}{3} \]
Thus, the selected response:
f(x) = 5⋅3^x; f(−1) = 5/3