Check One-One / Onto

🎥 Video Explanation

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📝 Question

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

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

where \([x]\) is the greatest integer ≤ \(x\).

• (a) many-one and onto
• (b) many-one and into
• (c) one-one and into
• (d) one-one and onto

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✅ Solution

🔹 Step 1: Use Property

For any real \(x\):

\[ [x+1]=[x]+1 \] —

🔹 Step 2: Simplify

\[ f(x)=[x]^2 + ([x]+1) – 3 \]

\[ f(x)=[x]^2 + [x] – 2 \] —

🔹 Step 3: Let \(n=[x]\)

Then:

\[ f(x)=n^2 + n – 2 \]

where \(n \in \mathbb{Z}\)

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🔹 Step 4: Check Injective

For all \(x \in [n, n+1)\), value of \(f(x)\) is same.

Multiple inputs → same output ⇒ ❌ Not one-one

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🔹 Step 5: Check Onto

Range consists of values:

\[ n^2 + n – 2, \quad n \in \mathbb{Z} \]

This does NOT cover all real numbers.

❌ Not onto

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🔹 Final Answer

\[ \boxed{\text{Option (b): many-one and into}} \]