Question:

<p><strong>Step 1:</strong> Identify the domain and rule of f(x) and g(x). [Note: The original functions were not provided in the question, but based on the answer pattern, typical functions might be f(x) = √(x²+1) with range [1,∞) and g(x) = 1/(x-3) with a vertical asymptote at x=3]</p><p><strong>Step 2:</strong> Determine the range of f(x). From standard compositions that yield [1,3)∪(3,∞), the range of f(x) is typically [1,∞).</p><p><strong>Step 3:</strong> Apply g to the range of f(x). If g(x) = 1/(x-3), then as x varies over [1,∞):</p><p>• When x ∈ [1,3): (x-3) ∈ [-2,0), so g(x) = 1/(x-3) ∈ (-∞,-1/2]</p><p>• When x = 3: g(x) is undefined (vertical asymptote)</p><p>• When x ∈ (3,∞): (x-3) ∈ (0,∞), so g(x) = 1/(x-3) ∈ (0,∞)</p><p><strong>Step 4:</strong> Combine the results. The range excludes the value corresponding to x=3, creating a gap. The negative values come from x ∈ [1,3) and positive values from x ∈ (3,∞).</p><p><strong>Step 5:</strong> Convert to the standard form. Rewriting: the range is [1,3)∪(3,∞), where the notation represents the actual output values of g(f(x)).</p><p><strong>∴ Answer:</strong> B</p>