Check Which Functions are Bijective

🎥 Video Explanation

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📝 Question

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

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

• A. \(f(x)=\frac{x}{2}\)
• B. \(g(x)=\sin\left(\frac{\pi x}{2}\right)\)
• C. \(h(x)=|x|\)
• D. \(k(x)=x^2\)

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✅ Solution

🔹 Option A: \(f(x)=\frac{x}{2}\)

Range: \[ [-1/2, 1/2] \]

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

❌ Not onto ⇒ Not bijective

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

Function is strictly increasing on \([-1,1]\)

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

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

✔️ One-one and onto ⇒ Bijective

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🔹 Option C: \(h(x)=|x|\)

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

❌ Not one-one

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🔹 Option D: \(k(x)=x^2\)

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

❌ Not one-one

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🔹 Final Answer

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