Question: A climate policy analyst models seasonal temperature variations with the function $ f(x) = 5\cos\left(\frac{\pi}{6}x\right) + 10 $. Determine the range of $ f(x) $ as $ x $ varies over all real numbers.

["Understanding the Seasonal Temperature Model: Range of the Function $ f(x) = 5\cos\left(\frac{\pi}{6}x\right) + 10 $", "Modeling seasonal temperature variations is essential for climate analysis and environmental planning. A climate policy analyst might use mathematical functions to represent and predict temperature trends over the year. One such function, frequently used in periodic modeling, is:", "$$\nf(x) = 5\cos\left(\frac{\pi}{6}x\right) + 10\n$$", "where $ x $ represents time (e.g., months or seasons), and the function captures the cyclical nature of temperature changes. In this article, we explore the range of this function and explain how its components contribute to seasonal temperatures.", "### Analyzing the Components of the Function", "The function is composed of a cosine term and a constant shift:", "- The cosine term: $ 5\cos\left(\frac{\pi}{6}x\right) $\n - Amplitude: 5\n - Period: $ \frac{2\pi}{\frac{\pi}{6}} = 12 $, meaning the temperature variation repeats every 12 months (approximately one year).\n - Range of $ \cos(\cdot) $ is $ [-1, 1] $, so $ 5\cos(\cdot) $ oscillates between $ -5 $ and $ 5 $.", "- Vertical shift: $ +10 $\n This shifts the entire cosine wave upward by 10 units.", "### Determining the Range", "Given the base oscillation $ -5 \leq 5\cos\left(\frac{\pi}{6}x\right) \leq 5 $, adding 10 gives:", "$$\n5\cos\left(\frac{\pi}{6}x\right) + 10 \in [-5 + 10,; 5 + 10] = [5,; 15]\n$$", "Thus, the minimum value of $ f(x) $ is $ 5 $, and the maximum is $ 15 $.", "### Interpretation in a Climate Context", "In practical terms, this function models seasonal temperature patterns where:", "- The peak temperature reaches 15°C during warmer periods (e.g., summer months),\n- The lowest temperature is 5°C, indicating cooler periods (e.g., winter months),\n- The average temperature remains stable at 10°C due to the vertical shift.", "Understanding this range helps analysts assess how climate policies might influence or stabilize seasonal extremes and supports informed decision-making in climate adaptation strategies.", "### Conclusion", "The function $ f(x) = 5\cos\left(\frac{\pi}{6}x\right) + 10 $ effectively models seasonal temperature fluctuations, with a clear and predictable range from $ 5 $ to $ 15 $. This insight enables climate scientists and policy makers to anticipate seasonal variation bounds and develop targeted environmental interventions.", "By mastering such mathematical models, analysts can enhance predictive accuracy and contribute to sustainable climate policy development."]