Limits, Continuity & Differentiability
Continuity of Composite Functions
Grade 12
<p>Given the function \( f(x) = \dfrac{1}{1-x} \). The points of discontinuity of the composite function, \( y = f(f(x)) \) are at \( x = 0 \)</p>
Step-by-Step Solution
Key Concept: A composite function f(f(x)) is discontinuous where either f(x) is discontinuous OR where f(x) takes a value at which f is discontinuous. For f(x) = 1/(1-x), find where f(x) is undefined AND where the inner function is undefined.
<p><strong>Step 1:</strong> Identify discontinuity of f(x) = 1/(1-x). This occurs at x = 1 (denominator = 0).</p><p><strong>Step 2:</strong> For f(f(x)), first find where f(x) is undefined: at x = 1, so f(f(x)) is discontinuous at x = 1.</p><p><strong>Step 3:</strong> Find where the output of f(x) equals 1 (the discontinuity point): f(x) = 1 ⟹ 1/(1-x) = 1 ⟹ 1 = 1-x ⟹ x = 0.</p><p><strong>Step 4:</strong> At x = 0: f(0) = 1/(1-0) = 1, so f(f(0)) = f(1) is undefined.</p><p><strong>Step 5:</strong> Therefore, y = f(f(x)) has discontinuities at x = 1 and x = 0.</p><p>∴ The statement that discontinuity occurs only at x = 0 is <strong>incomplete</strong>—discontinuities occur at both x = 0 and x = 1. If the question asks whether x = 0 is a point of discontinuity, the answer is <strong>B (True/Yes)</strong>.</p>
Correct Answer: B