The range of the function f : <span class="math-inline">\(\mathbb{N} \to \mathbb{Z}\)</span>; f(x) = <span class="math-inline">\((1-x)^{x-1}\)</span>, is -
Step-by-Step Solution
Key Concept: Since the domain is ℕ (natural numbers), we must evaluate f(x) = (1-x)^(x-1) for specific natural number values only. The range consists of all possible output values, which must be integers (codomain is ℤ).
<p><strong>Step 1:</strong> Identify the domain. The domain is ℕ (natural numbers). In standard convention, ℕ = {1, 2, 3, 4, ...}.</p><p><strong>Step 2:</strong> Evaluate f(x) = (1-x)^(x-1) at natural number values:</p><p><strong>For x = 1:</strong> f(1) = (1-1)^(1-1) = (0)^0. By convention, 0^0 = 1, so f(1) = 1</p><p><strong>For x = 2:</strong> f(2) = (1-2)^(2-1) = (-1)^1 = -1</p><p><strong>For x = 3:</strong> f(3) = (1-3)^(3-1) = (-2)^2 = 4</p><p><strong>For x = 4:</strong> f(4) = (1-4)^(4-1) = (-3)^3 = -27</p><p><strong>Step 3:</strong> Observe the pattern. For x ≥ 3, we get (1-x)^(x-1) where |1-x| ≥ 2 and the exponent grows. These values rapidly increase in magnitude and do not stabilize.</p><p><strong>Step 4:</strong> For x ≥ 3, the values are either large positive (even exponents) or large negative (odd exponents) numbers, none of which repeat the values -1 or 1.</p><p><strong>Step 5:</strong> The only values that appear in the range are f(1) = 1 and f(2) = -1. All other natural numbers produce values outside {-1, 1}.</p><p><strong>∴ Answer:</strong> B</p>
Correct Answer: B