Sets, Relations & Functions
General
Grade None
Question:
If \( f(x) + 2f\left(\frac{1}{x}\right) = 3x, x \neq 0, \) and S = \( \{x \in \mathbb{R} : f(x) = f(-x)\} \); then S :
contains more than two elements.
is an empty set.
contains exactly one element
contains exactly two elements
Step-by-Step Solution
<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: 4