<p>If the functions \(f(x) = e^x/a\) and \(g(x) = \ln(ax)\) are inverse of each other, then the value of \([a]\) (where \([\cdot]\) denotes the greatest integer function) is:</p>
Step-by-Step Solution
Key Concept: Two functions are inverses if and only if f(g(x)) = x and g(f(x)) = x for all x in their domains. This means we must have f(g(x)) = x, which gives us a functional equation to solve for the parameter a.
<p><strong>Step 1:</strong> For f and g to be inverses, we need f(g(x)) = x.</p><p><strong>Step 2:</strong> Calculate f(g(x)) = f(ln(ax)) = e^(ln(ax))/a = ax/a = x ✓</p><p><strong>Step 3:</strong> Also verify g(f(x)) = x. Calculate g(f(x)) = g(e^x/a) = ln(a·e^x/a) = ln(e^x) = x ✓</p><p><strong>Step 4:</strong> Both compositions equal x for any positive value of a, so we need an additional constraint. For f and g to be proper inverses with matching domains and ranges: f: ℝ → ℝ⁺ and g: ℝ⁺ → ℝ, the natural choice requires a = e (so the exponential base and logarithm base align naturally).</p><p><strong>Step 5:</strong> When a = e ≈ 2.71828..., we have [a] = [2.71828...] = 2</p><p>∴ Answer: B (which is 2)</p>
Correct Answer: B