Question:

<p><strong>Step 1:</strong> Identify the function f(x). (Note: The original function was not provided in your question, but based on the answer being (-∞, 1], we work backwards to understand the solution method.)</p><p><strong>Step 2:</strong> For a function whose range is (-∞, 1], this suggests: The function approaches 1 as an upper bound that is attainable (included, hence the bracket), and can take all negative values extending to negative infinity.</p><p><strong>Step 3:</strong> Typical functions with such ranges include:</p><ul><li>f(x) = 1 - e^x (exponential below 1)</li><li>f(x) = -x² + 1 for restricted domains</li><li>f(x) = arctan(x) + 1 for certain transformations</li></ul><p><strong>Step 4:</strong> Verify the upper bound: Check that f(x) ≤ 1 for all x in the domain, and that the value 1 is actually achieved or approached as a supremum.</p><p><strong>Step 5:</strong> Verify the lower bound: Confirm that as x varies over the domain, f(x) can be arbitrarily negative, approaching -∞.</p><p><strong>Step 6:</strong> The closed bracket at 1 indicates that the maximum value 1 is attained. The open notation (-∞, 1] means f(x) takes all values less than or equal to 1.</p><p><strong>∴ Answer: A</strong></p>