Sets, Relations & Functions
General
Grade None
Question:
<p>If <span class="math-inline">\(f(x)=\log\dfrac{1-x}{1+x}\)</span>, <span class="math-inline">\(|x|<1\)</span>, find <span class="math-inline">\(f\!\left(\dfrac{2x}{1+x^2}\right)\)</span>.</p>
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>Put u=2x/(1+x²). 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><div class="key-concept"><strong>Key Concept:</strong> <span class="math-inline">\(\frac{2x}{1+x^2}\)</span> substitution → square-ratio identity</div></div>
Correct Answer: 2f(x)