Question:

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