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

📺 Video Explanation

📝 Question

Check whether the function

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

is:

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

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✅ Solution

🔹 Step 1: Check Injection (One-One)

Assume:

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

Then:

\[ x_1^2=x_2^2 \]

Since \(x_1,x_2\in\mathbb{N}\), both are positive.

So:

\[ x_1=x_2 \]

✔ Hence, function is one-one.

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🔹 Step 2: Check Surjection (Onto)

For onto, every natural number must have pre-image.

But:

\[ 2\in\mathbb{N} \]

There is no natural number \(x\) such that:

\[ x^2=2 \]

So:

\[ 2 \] is not in range.

❌ Hence, function is not onto.

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🎯 Final Answer

\[ \boxed{\text{f is one-one but not onto}} \]

So:

✔ Injection
❌ Surjection
❌ Bijection

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🚀 Exam Shortcut

• Square function increases on natural numbers → one-one
• Only perfect squares appear in range
• Missing numbers mean not onto