Check Function \(f(x)=x^2\) on \( \mathbb{Z} \)

📺 Video Explanation

📝 Question

Check whether the function

\[ f:\mathbb{Z}\to\mathbb{Z},\quad f(x)=x^2 \]

is:

• injection (one-one)
• surjection (onto)
• bijection

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✅ Solution

🔹 Step 1: Check Injection (One-One)

A function is one-one if:

\[ f(x_1)=f(x_2)\Rightarrow x_1=x_2 \]

Take:

\[ x_1=2,\quad x_2=-2 \]

Then:

\[ f(2)=4,\quad f(-2)=4 \]

But:

\[ 2\neq-2 \]

❌ Not one-one.

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🔹 Step 2: Check Surjection (Onto)

For onto, every integer must have pre-image.

But:

\[ -1\in\mathbb{Z} \]

There is no integer \(x\) such that:

\[ x^2=-1 \]

So:

\[ -1 \] is not in range.

❌ Not onto.

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🎯 Final Answer

\[ \boxed{\text{f is neither one-one nor onto}} \]

So:

❌ Injection
❌ Surjection
❌ Bijection

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🚀 Exam Shortcut

• \(x^2\) gives same value for \(x\) and \(-x\)
• Negative integers are never obtained
• So neither injective nor surjective