To determine which function best models the provided data and to calculate the value of the function at \( x = -1 \), we can analyze the given options.
The provided data points are:
- \( (0, 5) \)
- \( (1, 15) \)
- \( (2, 45) \)
- \( (3, 135) \)
- \( (4, 405) \)
Let's evaluate the options one by one:
-
Option 1: \( f(x) = 41.43x^2 - 73.71x + 19.86 \)
- Let's test \( f(-1) \): \[ f(-1) = 41.43(-1)^2 - 73.71(-1) + 19.86 = 41.43 + 73.71 + 19.86 \approx 135 \quad (\text{Correct calculation}) \]
-
Option 2: \( f(x) = 5 \cdot 3^x \)
- Let's test \( f(-1) \): \[ f(-1) = 5 \cdot 3^{-1} = 5 \cdot \frac{1}{3} = \frac{5}{3} \approx 1.67 \]
-
Option 3: \( f(x) = 2.23 \cdot (1.40)^x \)
- Let's test \( f(-1) \): \[ f(-1) = 2.23 \cdot (1.40)^{-1} = 2.23 \cdot \frac{1}{1.40} \approx 1.59 \]
-
Option 4: \( f(x) = 92x - 63 \)
- Let's test \( f(-1) \): \[ f(-1) = 92(-1) - 63 = -92 - 63 = -155 \]
Next, we look at which function fits the given data points best. Analyzing the outputs as given, we notice:
- The first function gives \( f(-1) = 135 \), which does not match how values typically function.
- The second option, when \( x = -1 \), gives a value of approximately \( 1.67\).
- The third option gives \( f(-1) \approx 1.59\).
- The fourth option correctly gives \( f(-1) = -155 \).
In the context of potential modeling, the polynomial function \( f(x) = 41.43x^2 - 73.71x + 19.86 \) gives a critical fit for points mentioned for non-negative integers. Thus, the best choice of the model from a dataset perspective, specifically focusing on defining \( f(-1)\) to find its own functionalities when seeking insights into escalating patterns, retains remarkable contribution value.
Conclusively, among these formulations, the most fitting across formal selection would identify with \( f(-1) = 135 \) as quintessentially representing the polynomial expansion approach.
Final Answer:
Option 1: \( f(x) = 41.43x^2 - 73.71x + 19.86 \); \( f(-1) = 135 \).