To find a function that best models the data you've provided, let's look at the relationship between the \(x\) and \(y\) values:
- \( (0, 5) \)
- \( (1, 15) \)
- \( (2, 45) \)
- \( (3, 135) \)
- \( (4, 405) \)
A potential approach to model the data is to look for a polynomial or exponential function. Given the rapid increase in \(y\) values, an exponential or a polynomial function might be present.
We can try dividing the \(y\) values by the previous ones to check for a pattern:
- \( \frac{15}{5} = 3 \)
- \( \frac{45}{15} = 3 \)
- \( \frac{135}{45} = 3 \)
- \( \frac{405}{135} = 3 \)
It appears that each subsequent \(y\) value is \(3\) times the previous one after \(x = 0\). This suggests a recursive relationship or possibly a direct function.
The \(y\) values can be expressed as:
- \(y(0) = 5\)
- \(y(1) = 5 \cdot 3^1 = 15\)
- \(y(2) = 5 \cdot 3^2 = 45\)
- \(y(3) = 5 \cdot 3^3 = 135\)
- \(y(4) = 5 \cdot 3^4 = 405\)
From this pattern, we can derive a formula for \(y\) as: \[ y = 5 \cdot 3^x \]
Now, to find the value of the function at \(x = -1\): \[ y(-1) = 5 \cdot 3^{-1} = 5 \cdot \frac{1}{3} = \frac{5}{3} \]
Thus, the value of the function at \(x = -1\) is: \[ \frac{5}{3} \text{ or } 1.67 \text{ (approximately).} \]