To determine the function that best models the data and find the value of the function at \( x = -1 \), we first analyze the given data:
- \( f(0) = 5 \)
- \( f(1) = 15 \)
- \( f(2) = 45 \)
- \( f(3) = 135 \)
- \( f(4) = 405 \)
Looking at the y-values, we can see they seem to increase rapidly. This suggests a potential exponential relationship.
Let's evaluate the candidate functions one by one:
-
Candidate: \( f(x) = 5 \cdot 3^x \)
- \( f(0) = 5 \cdot 3^0 = 5 \)
- \( f(1) = 5 \cdot 3^1 = 15 \)
- \( f(2) = 5 \cdot 3^2 = 45 \)
- \( f(3) = 5 \cdot 3^3 = 135 \)
- \( f(4) = 5 \cdot 3^4 = 405 \)
- This function matches all given points.
To find \( f(-1) \): \[ f(-1) = 5 \cdot 3^{-1} = 5 \cdot \frac{1}{3} = \frac{5}{3} \]
-
Candidate: \( f(x) = 41.43x^2 - 73.71x + 19.86 \)
- You can evaluate this polynomial, but we noticed it doesn't seem to match the growth pattern.
-
Candidate: \( f(x) = 2.23 \cdot (1.40)^x \)
- Evaluating it yields values that don’t match the rapid growth.
-
Candidate: \( f(x) = 92x - 63 \)
- This is a linear function, which will not match the exponential growth either.
From the evaluation, the best model for the data is \( f(x) = 5 \cdot 3^x \).
Thus, the value of the function at \( x = -1 \) is: \[ f(-1) = \frac{5}{3} \]
Final Answer: \( f(x) = 5 \cdot 3^x \); \( f(-1) = \frac{5}{3} \)