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

📺 Video Explanation

📝 Question

Check whether the function

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

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:

\[ x=1,\quad x=-2 \]

Then:

\[ f(1)=1^2+1=2 \]

\[ f(-2)=(-2)^2+(-2)=4-2=2 \]

But:

\[ 1\neq-2 \]

❌ Not one-one.

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🔹 Step 2: Check Surjection (Onto)

For onto, every integer must have pre-image.

Take:

\[ y=1 \]

Need:

\[ x^2+x=1 \]

or:

\[ x^2+x-1=0 \]

This equation has no integer solution.

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

• Try small values to test injection
• Find missing codomain value to test onto
• Quadratic functions on integers usually fail bijection