- Examine each given interval to check if the function values \(f(x)\) consistently increase as \(x\) increases.
- In the interval \((-6, -3)\), the values go from \(34\) to \(-11\), showing a decrease.
- In the interval \((-3, -1)\), the values go from \(-11\) to \(-6\) to \(-1\), which is a strictly increasing sequence.
- In the interval \((-3, 0)\), the values increase then decrease at \(x=0\).
- In the interval \((-6, -5)\), the values go from \(34\) to \(3\), showing a decrease.
- The correct interval is \(\boxed{(-3, -1)}\).
Explanation
- Understanding Increasing Functions and Data Analysis
To determine the interval where the function \(f(x)\) is increasing, we look at the provided table of values:
| \(x\) | \(f(x)\) \) -6 | 34 \) -5 | 3 \) -4 | -10 \) -3 | -11 \) -2 | -6 \) -1 | -1 \) 0 | -2 \) 1 | -15 \) |
A function \(f(x)\) is considered to be increasing on an interval if, for any two values \(x_1\) and \(x_2\) within that interval such that \(x_1 < x_2\), the corresponding functional values satisfy \(f(x_1) < f(x_2)\). We will analyze the behavior of the function across the given options by comparing the values at consecutive \(x\)-values.
- Evaluating the Intervals
We examine each proposed interval by looking at the values of \(f(x)\) at the specified \(x\)-coordinates:
- Option \((-6, -3)\): The values are \(f(-6) = 34\), \(f(-5) = 3\), \(f(-4) = -10\), and \(f(-3) = -11\). Since \(34 > 3 > -10 > -11\), the function values are decreasing.
- Option \((-3, -1)\): The values are \(f(-3) = -11\), \(f(-2) = -6\), and \(f(-1) = -1\). We compare consecutive values:
- From \(x = -3\) to \(x = -2\), the function increases from \(-11\) to \(-6\) (since \(-11 < -6\)).
- From \(x = -2\) to \(x = -1\), the function increases from \(-6\) to \(-1\) (since \(-6 < -1\)).
Since the values consistently increase as \(x\) increases, the function is increasing on the interval \((-3, -1)\).
- Option \((-3, 0)\): The values are \(f(-3) = -11\), \(f(-2) = -6\), \(f(-1) = -1\), and \(f(0) = -2\). While the function increases from \(x = -3\) to \(x = -1\), it then decreases from \(f(-1) = -1\) to \(f(0) = -2\). Thus, it is not increasing on the entire interval \((-3, 0)\).
- Option \((-6, -5)\): The values are \(f(-6) = 34\) and \(f(-5) = 3\). Since \(34 > 3\), the function is decreasing.
- Conclusion
Based on our analysis, the only interval where the function values strictly increase for the given points is \((-3, -1)\). Therefore, the correct interval is \((-3, -1)\).
\(\boxed{(-3, -1)}\)
Examples
In real life, identifying intervals where a function increases or decreases is essential for tracking trends. For instance, if you were analyzing the temperature changes during the night, the table might show the temperature at specific hours. By looking at the data, you could identify the exact time window when the temperature was rising, which is helpful for predicting weather patterns or managing energy usage for heating. The mathematical concept of an increasing function is the foundation for such trend analysis.