Check Function \(f(x)=x^3\) on \( \mathbb{Z} \)

📺 Video Explanation

📝 Question

Check whether the function

\[ f:\mathbb{Z}\to\mathbb{Z},\quad f(x)=x^3 \]

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:

\[ x_1^3=x_2^3 \]

Cube function is strictly increasing on integers.

So:

\[ x_1=x_2 \]

✔ Hence, function is one-one.

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🔹 Step 2: Check Surjection (Onto)

For onto, every integer must have pre-image.

But:

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

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

\[ x^3=2 \]

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

• Cube function is strictly increasing → injective
• Only perfect cubes appear in range
• Missing integers mean not onto