Check Function \(f(x)=|x|\) on \( \mathbb{R} \)

📺 Video Explanation

📝 Question

Check whether the function

\[ f:\mathbb{R}\to\mathbb{R},\quad f(x)=|x| \]

is:

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

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✅ Solution

🔹 Step 1: Check Injection (One-One)

A function is one-one if different inputs have different outputs.

Take:

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

Then:

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

But:

\[ 2\neq-2 \]

Different inputs give same output.

❌ Not one-one.

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🔹 Step 2: Check Surjection (Onto)

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

But:

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

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

\[ |x|=-1 \]

because modulus is always non-negative.

❌ 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

• Absolute value gives same result for \(x\) and \(-x\)
• Negative values are never in range
• So neither injective nor surjective