- Analyze the sign of the function values from the provided table, noting that \(f(x) < 0\) for \(x = -8, -6\).
- Recognize that \(f(-4) = 0\) and that the function values increase as \(x\) increases.
- Conclude that the function is negative for all \(x\) values strictly less than \(-4\), which corresponds to the interval \((-\infty, -4)\).
- The final answer is \(\boxed{(-\infty, -4)}\).
Explanation
- Analyzing the Function Values
The problem asks for the interval over which the function \(f(x)\) is negative based on the provided table of values. We are given the following points:
\[ \begin{array}{|l|l|} \ x & f(x) \\ -8 & -16 \\ -6 & -8 \\ -4 & 0 \\ -2 & 8 \\ 0 & 16 \\ 2 & 32 \\ 4 & 64 \\ 6 & 128 \\ \end{array} \]
By observing the table, we see that for \(x = -8\) and \(x = -6\), the function values are \(-16\) and \(-8\), respectively, both of which are less than \(0\). At \(x = -4\), the function value is \(0\). For all values of \(x\) greater than \(-4\) (i.e., \(-2, 0, 2, 4, 6\)), the function values are all positive.
- Determining the Interval
Given the trend in the table, the function \(f(x)\) appears to be monotonically increasing. As \(x\) increases, \(f(x)\) consistently increases. Since \(f(-4) = 0\) and the function is increasing, it follows that for any \(x\) less than \(-4\), the function value \(f(x)\) must be less than \(0\), and for any \(x\) greater than \(-4\), the function value \(f(x)\) must be greater than \(0\). Therefore, the entire interval over which the function \(f(x)\) is negative is \((-\infty, -4)\).
- Conclusion
Comparing our result \((-\infty, -4)\) with the given options, it matches the fourth choice exactly. Thus, the function is negative on the interval \((-\infty, -4)\). The final answer is \(\boxed{(-\infty, -4)}\).
Examples
Consider a situation where you are tracking the temperature inside a freezer over time. If your recorded data shows that the temperature is negative at early time intervals but reaches exactly \(0^\circ\text{C}\) at a specific time \(t_0\), and then begins to rise into positive values for all times \(t > t_0\), you can determine the exact period during which the freezer was below freezing. This type of data analysis is essential for maintaining safety standards in food storage and chemical laboratories, ensuring that conditions remain within the necessary ranges.