Check One-One / Onto

🎥 Video Explanation

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📝 Question

Let \( f:\mathbb{R} \to \mathbb{R} \) be defined by

\[ f(x)=(x-1)(x-2)(x-3) \]

• A. one-one but not onto
• B. onto but not one-one
• C. both one-one and onto
• D. neither one-one nor onto

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✅ Solution

🔹 Step 1: Nature of Function

\(f(x)\) is a cubic polynomial.

As \(x \to \infty\), \(f(x) \to \infty\)
As \(x \to -\infty\), \(f(x) \to -\infty\)

✔️ Covers all real values ⇒ Onto

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🔹 Step 2: Check Injective

Cubic with turning points ⇒ not strictly monotonic.

Example:

\[ f(1)=0,\quad f(2)=0,\quad f(3)=0 \]

Different inputs → same output ⇒ ❌ Not one-one

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🔹 Final Answer

\[ \boxed{\text{Option B: onto but not one-one}} \]