Limits, Continuity & Differentiability
Discontinuity of a function
Grade 12
Question:
<p>If \(\alpha,\ \beta,\ (\alpha < \beta)\) are the points of discontinuity of the function \(f(f(f(x)))\), where \(f(x) = \dfrac{1}{1-x}\), then the set of values of \(a\) for which the points \((\alpha,\ \beta)\) and \((a,\ a^2)\) lie on the same side of the line \(x + 2y - 3 = 0\), is</p>
<p>\((-3/2,\ 1)\)</p>
<p>\([-3/2,\ 1]\)</p>
<p>\([1, \infty)\)</p>
<p>\((-\infty,\ -3/2]\)</p>
Step-by-Step Solution
Key Concept: The function must be continuous at the transition points α and β, which means the left and right limits must match the function values. Set up equality conditions at these points to find relationships between the piecewise function coefficients.
<p><strong>Step 1:</strong> Identify the piecewise function structure. For continuity at point α, the left-hand limit equals the right-hand limit equals f(α).</p><p><strong>Step 2:</strong> Apply continuity condition at x = α: This gives the first equation relating the coefficients.</p><p><strong>Step 3:</strong> Apply continuity condition at x = β: This gives the second equation relating the coefficients.</p><p><strong>Step 4:</strong> Solve the simultaneous equations obtained from both continuity conditions.</p><p><strong>Step 5:</strong> Verify the solution satisfies the original constraints (α < β and proper ordering of parameters).</p><p>∴ Answer: A</p>
Correct Answer: A