Check Which Functions are Bijective

🎥 Video Explanation

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📝 Question

Let \(A=\{x\in\mathbb{R}:-1\le x\le 1\}\).

Which of the following functions \(A \to A\) are bijections?

• A. \(f(x)=|x|\)
• B. \(f(x)=\sin\left(\frac{\pi x}{2}\right)\)
• C. \(f(x)=\sin\left(\frac{\pi x}{4}\right)\)
• D. none of these

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✅ Solution

🔹 Option A: \(f(x)=|x|\)

\[ f(-x)=f(x) \]

❌ Not one-one ⇒ Not bijective

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🔹 Option B: \(f(x)=\sin\left(\frac{\pi x}{2}\right)\)

Strictly increasing on \([-1,1]\)

\[ f(-1)=-1,\quad f(1)=1 \]

Range: \[ [-1,1] \]

✔️ One-one and onto ⇒ Bijective

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🔹 Option C: \(f(x)=\sin\left(\frac{\pi x}{4}\right)\)

Range:

\[ \left[-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right] \]

Not equal to \([-1,1]\)

❌ Not onto ⇒ Not bijective

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🔹 Final Answer

\[ \boxed{\text{Option B}} \]