<p>For \(f(x)=e^x\), \(g(x)=\sin^{-1}x\), which are necessarily true?</p>
Step-by-Step Solution
<div class="solution"><p><strong>Step 1:</strong> Domain of $g\circ f$: need $e^x\in[-1,1]$. Since $e^x>0$, need $e^x\le 1\implies x\le 0$. Domain = $(-\infty,0]$ ✓ (C true, A false).</p><p><strong>Step 2:</strong> Range of $g\circ f$: $e^x\in(0,1]$ \to $\sin^{-1}(e^x)\in(0,\pi/2]$. This is a subset of range of g = $[-\pi/2,\pi/2]$ ✓ (B true).</p><p><strong>Step 3:</strong> Range is $(0,\pi/2]$, not $[-\pi/2,0]$ (D false).</p><p><strong>Answer: (B),(C)</strong></p><div class="trap-box"><strong>Trap:</strong> Forgetting eˣ>0 always, so the sin⁻^1 output is always positive.<div class="key-concept"><strong>Key Concept:</strong> Range of inner exponential restricts which branch of outer sin⁻^1 is accessed
Correct Answer: B,C