- Observe the function values in the table: \(f(-2)=0\) and \(f(1)=0\), indicating roots at \(x=-2\) and \(x=1\).
- Note that for \(x\) values between \(-2\) and \(1\), specifically \(x=-1\) and \(x=0\), the values \(f(x)=2\) are positive.
- Observe that for \(x < -2\) and \(x > 1\), the function values are negative.
- Conclude that the interval over which the function is positive is \((-2, 1)\), which corresponds to the second option.
Final Answer: \(\boxed{(-2, 1)}\)
Explanation
- Analyzing the Function Values
To determine the interval over which the function \(f(x)\) is positive, we examine the provided table of values for different \(x\):
| \(x\) | \(f(x)\) |
| :--- | :--- |
| \(-3\) | \(-2\) |
| \(-2\) | \(0\) |
| \(-1\) | \(2\) |
| \(0\) | \(2\) |
| \(1\) | \(0\) |
| \(2\) | \(-8\) |
| \(3\) | \(-10\) |
| \(4\) | \(-20\) |
By inspecting these values, we can determine the sign of the function at various points. For \(x < -2\), such as \(x = -3\), \(f(x) = -2\), so the function is negative. At \(x = -2\), \(f(x) = 0\), identifying a root of the function. For \(x\) values between \(-2\) and \(1\), specifically \(x = -1\) and \(x = 0\), we see that \(f(-1) = 2\) and \(f(0) = 2\), indicating the function is positive in this range. At \(x = 1\), \(f(x) = 0\), which is another root. Finally, for \(x > 1\), such as \(x = 2, 3, 4\), the values \(f(x)\) are \(-8, -10, -20\), showing the function is negative again.
- Selecting the Correct Interval
Given the analysis that \(f(x) = 0\) at \(x = -2\) and \(x = 1\), and \(f(x) > 0\) for values between these two points, the interval over which the function is positive is \((-2, 1)\). We can now evaluate the given options:
- \((-\infty, 1)\): This interval includes points where \(x < -2\), but we know \(f(x) < 0\) in that region.
- \((-2, 1)\): This matches our finding where the function values are positive.
- \((-\infty, 0)\): This interval includes points where \(x < -2\), where the function is negative.
- \((1, 4)\): In this interval, the function values are negative.
Comparing our derived interval with the options, \((-2, 1)\) is the correct choice.
Examples
Understanding functions through data tables is useful in many real-life applications. For example, consider an engineer monitoring the temperature of a chemical reaction. They might record temperature readings at specific time intervals. If the temperature is defined as positive when it is above a certain baseline (e.g., above room temperature), identifying the time interval where the temperature is positive helps them understand the duration and peak intensity of the reaction. This analysis allows for precise control over manufacturing processes or safety protocols.