If x^4 f\left( x \right) - \sqrt{1 - \sin 2\pi x} = |f\left( x \right)| - 2f\left( x \right), then f\left( -2 \right) equals
Step-by-Step Solution
Key Concept: f and f⁻¹ intersect on y=x. Solve f(x)=x.
<div class="solution"><p><strong>Key Idea:</strong> f and f⁻¹ intersect on y=x. Solve f(x)=x.</p><p><strong>Step 1:</strong> <span class="math-inline">$x = 1+\frac{3}{2}\sqrt{x}$</span>. Let <span class="math-inline">$t=\sqrt{x}>0$</span>.</p><p><strong>Step 2:</strong> <span class="math-inline">$2t^2-3t-2=0 \implies (2t+1)(t-2)=0 \implies t=2$</span></p><p><strong>Step 3:</strong> <span class="math-inline">$x=4$</span>, intersection point is <span class="math-inline">$(4,4)$</span></p><div class="trap-box"><strong>Trap:</strong> Writing the inverse first is slower. Use symmetry line y=x directly.</div><div class="key-concept"><strong>Key Concept:</strong> f and f⁻¹ intersect on y=x</div></div>
Correct Answer: A