Sets, Relations & Functions
Grade 11
Question:
<p>If \(f(x)=\log\dfrac{1-x}{1+x}\), \(|x|<1\), find \(f\!\left(\dfrac{2x}{1+x^2}\right)\).</p>
Step-by-Step Solution
<div class="solution"><p>Put u=2x/(1+x^2). Then (1-u)/(1+u)=((1-x)/(1+x))^2. So f(u)=2log((1-x)/(1+x))=2f(x).</p><p><strong>Answer: 2f(x)</strong></p><div class="trap-box"><strong>Trap:</strong> Don't expand log early. First simplify inner fraction into perfect square.<div class="key-concept"><strong>Key Concept:</strong> $\frac{2x}{1+x^2}$ substitution \to square-ratio identity
Correct Answer: 2f(x)