Check Which Functions are Bijective

🎥 Video Explanation

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📝 Question

Which of the following functions \(f:\mathbb{Z} \to \mathbb{Z}\) are bijections?

• A. \(f(x)=x^3\)
• B. \(f(x)=x+2\)
• C. \(f(x)=2x+1\)
• D. \(f(x)=x^2+x\)

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✅ Solution

🔹 Option A: \(f(x)=x^3\)

Injective ✔️ (strictly increasing)

Not onto ❌ (e.g., 2 is not a cube of any integer)

⇒ Not bijective

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🔹 Option B: \(f(x)=x+2\)

Injective ✔️ (linear function)

Onto ✔️ (for any \(y\), \(x=y-2\))

⇒ Bijective

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🔹 Option C: \(f(x)=2x+1\)

Injective ✔️

Not onto ❌ (only odd numbers are obtained)

⇒ Not bijective

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🔹 Option D: \(f(x)=x^2+x\)

Not injective ❌ (e.g., \(f(0)=0\), \(f(-1)=0\))

Not onto ❌

⇒ Not bijective

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🔹 Final Answer

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