Relations & Functions
Functions and Greatest Integer Function
Grade 12
Question:
<p>If \(f(x) \in (1, 2]\), \([f(x)] = 1, 2\), \(\dfrac{2(k+1)}{3} = 3 \Rightarrow k = \dfrac{7}{2}\) and \(\dfrac{\mu}{3} = 2 \Rightarrow \mu = 6\), find the value of \(2k + \mu\).</p>
Step-by-Step Solution
Key Concept: When f(x) ∈ (1, 2], the greatest integer function [f(x)] takes values 1 (when 1 < f(x) ≤ 2). Use the given functional equations to solve for parameters k and μ independently, then compute the linear combination.
<p><strong>Step 1:</strong> Analyze the range constraint. Since f(x) ∈ (1, 2], we have [f(x)] = 1 (the greatest integer ≤ f(x) is 1 when 1 < f(x) ≤ 2).</p><p><strong>Step 2:</strong> Solve for k using the first equation: 2(k+1)/3 = 3</p><p>Multiply both sides by 3: 2(k+1) = 9</p><p>2k + 2 = 9</p><p>2k = 7</p><p>k = 7/2</p><p><strong>Step 3:</strong> Solve for μ using the second equation: μ/3 = 2</p><p>μ = 6</p><p><strong>Step 4:</strong> Calculate 2k + μ</p><p>2k + μ = 2(7/2) + 6 = 7 + 6 = 13</p><p>∴ Answer: 13</p>
Correct Answer: 13