Functions
General
Grade 12
Question:
A function f has domain [-1, 2] and range [0, 1]. The domain and range respectively of the function g defined by g(x) = 1 - f(x + 1) is
[-1, 1] ; [-1, 0]
[-2, 1]; [0, 1]
[0, 2]; [-1, 0]
[1, 3]; [-1, 0]
Step-by-Step Solution
Key Concept: Reciprocal substitution converts the recurrence into an AP.
<div class="solution"><p><strong>Key Idea:</strong> Reciprocal substitution converts the recurrence into an AP.</p><p><strong>Step 1:</strong> Let <span class="math-inline">$y_n = 1/x_n$</span>. Then <span class="math-inline">$y_{n+1} = y_n - 1$</span> (arithmetic progression).</p><p><strong>Step 2:</strong> <span class="math-block">$$x_n = \frac{\alpha}{1 - n\alpha}$$</span></p><p><strong>Step 3:</strong> Set <span class="math-inline">$x_{2011} = -1/2012$</span>: <span class="math-block">$$\frac{\alpha}{1-2011\alpha} = -\frac{1}{2012}$$</span>Cross-multiplying: <span class="math-inline">$2012\alpha = -1+2011\alpha \implies \alpha = -1$</span></p><p><strong>Answer: <span class="math-inline">$\alpha = -1$</span></strong></p><div class="trap-box"><strong>Trap:</strong> The sign matters completely. Missing the negative sign changes the answer to a non-integer value.</div><div class="key-concept"><strong>Key Concept:</strong> Fractional linear recurrence → reciprocal substitution → AP</div></div>
Correct Answer: B