Question:

<div class="solution"><p><strong>Key Idea:</strong> Solve for <span class="math-inline">$2^y$</span>. Exponential is always positive.</p><p><strong>Step 1:</strong> <span class="math-inline">$2^y = 2 - 2^x$</span>. For real <span class="math-inline">$y$</span>, need <span class="math-inline">$2^y > 0$</span>.</p><p><strong>Step 2:</strong> <span class="math-block">$$2 - 2^x > 0 \implies 2^x < 2 \implies x < 1$$</span></p><p><strong>Answer: <span class="math-inline">$(-\infty, 1)$</span></strong></p><div class="trap-box"><strong>Trap:</strong> At <span class="math-inline">$x=1$</span>, the equation gives <span class="math-inline">$2^y=0$</span> which is impossible. So <span class="math-inline">$x=1$</span> is not included.</div><div class="key-concept"><strong>Key Concept:</strong> Domain — exponential output is always strictly positive</div></div>