Check Injective / Surjective

🎥 Video Explanation

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📝 Question

Let \( f:\mathbb{R} \to \mathbb{R} \),

\[ f(x)=x^2 \]

• A. injective but not surjective
• B. surjective but not injective
• C. both injective and surjective
• D. neither injective nor surjective

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✅ Solution

🔹 Step 1: Check Injective

\[ f(x)=x^2 \Rightarrow f(2)=4,\; f(-2)=4 \]

Different inputs → same output ⇒ ❌ Not injective

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🔹 Step 2: Check Surjective

Range of \(f(x)=x^2\):

\[ [0,\infty) \]

Codomain is \(\mathbb{R}\)

Negative values not covered ⇒ ❌ Not surjective

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🔹 Final Answer

\[ \boxed{\text{Option D: neither injective nor surjective}} \]