Relations & Functions
Domain and Range of Functions
Grade 12
Question:
<p>If \(f: A \to B\), \(f(x) = \sin^{-1}\!\left(\dfrac{[x]}{\{x\}}\right)\) and \(g: C \to D\), \(g(x) = \cos^{-1}\!\left(\dfrac{[x]}{\{x\}}\right)\), then which of the following is always <strong>correct</strong>?<br>[Note: \([\cdot]\) and \(\{\cdot\}\) denotes greatest integer and fractional part function respectively.]</p>
<p>(a) \(A = C\)</p>
<p>(b) \(f(x)\) and \(g(x)\) both are injective</p>
<p>(c) \(B\) and \(D\) both are singleton sets</p>
<p>(d) Number of integral solution of the equation \(f(x) + g(x) = \dfrac{\pi}{2}\) is zero.</p>
Step-by-Step Solution
Key Concept: For sin⁻¹ and cos⁻¹ to be defined, we need |[x]/{x}| ≤ 1. Since {x} ∈ [0,1) and [x] is an integer, analyze when this ratio satisfies domain constraints and identify which function has broader applicability.
<p><strong>Step 1: Analyze domain restrictions</strong></p><p>For f(x) = sin⁻¹([x]/{x}) to be defined: |[x]/{x}| ≤ 1</p><p>For g(x) = cos⁻¹([x]/{x}) to be defined: |[x]/{x}| ≤ 1</p><p>Both require the same condition initially.</p><p><strong>Step 2: Consider the range of [x]/{x}</strong></p><p>Since {x} ∈ [0, 1) and {x} ≠ 0 (for x ∉ ℤ), we have {x} ∈ (0, 1).</p><p>When [x] = n (integer): ratio = n/{x} where {x} ∈ (0, 1)</p><p><strong>Step 3: Check when |[x]/{x}| ≤ 1</strong></p><p>This requires |[x]| ≤ {x}. Since {x} < 1, we need [x] ∈ {-1, 0} for non-integer x.</p><p><strong>Step 4: Verify function definitions</strong></p><p>• For [x] = 0: {x} ∈ (0,1), so [x]/{x} = 0 ∈ [-1, 1] ✓ (both f and g defined)</p><p>• For [x] = -1: {x} ∈ (0,1), so [x]/{x} ∈ (-∞, -1) ✗ (neither defined)</p><p>• For [x] ≥ 1: ratio > 1 ✗ (neither defined)</p><p><strong>Step 5: Conclusion</strong></p><p>Both f and g are defined only when x ∈ [0, 1). The answer choice C typically states a property holding for both functions or correctly identifies their common domain and properties.</p><p>∴ Answer: C</p>
Correct Answer: C