Find Set \(A\) for Bijective Function \(f(x)=\sin x\)

📝 Question

Let:

\[ f:\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\to A,\quad f(x)=\sin x \]

If \(f\) is bijective, find the set \(A\).

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✅ Solution

🔹 Step 1: Check injectivity

\(\sin x\) is strictly increasing on \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\).

Hence, \(f\) is one-one.

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🔹 Step 2: Find range

We know:

\[ \sin\left(-\frac{\pi}{2}\right)=-1,\quad \sin\left(\frac{\pi}{2}\right)=1 \]

But endpoints are not included.

So values lie between \(-1\) and \(1\), but do not include them:

:contentReference[oaicite:0]{index=0} —

🔹 Step 3: Determine set \(A\)

For bijection, codomain must be equal to range.

Thus,

\[ A=(-1,1) \] —

🎯 Final Answer

\[ \boxed{A=(-1,1)} \]

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🚀 Exam Shortcut

• For bijection ⇒ codomain = range
• \(\sin x\) on this interval ⇒ \((-1,1)\)
• Endpoints excluded ⇒ open interval