Check Function \(f(x)=1+x^2\) on \( \mathbb{R} \)

📺 Video Explanation

📝 Question

Check whether the function

\[ f:\mathbb{R}\to\mathbb{R},\quad f(x)=1+x^2 \]

is:

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

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✅ Solution

🔹 Step 1: Check Injection (One-One)

Take:

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

Then:

\[ f(2)=1+4=5,\quad f(-2)=1+4=5 \]

But:

\[ 2\neq-2 \]

❌ Not one-one.

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🔹 Step 2: Check Surjection (Onto)

Since:

\[ x^2\geq0 \]

Therefore:

\[ 1+x^2\geq1 \]

So range is:

\[ [1,\infty) \]

But codomain is:

\[ \mathbb{R} \]

Numbers like:

\[ 0,\ -2 \]

are 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

• Square terms repeat values → not injective
• Addition shifts range upward
• Compare range with codomain carefully