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

📺 Video Explanation

📝 Question

Check whether the function

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

is:

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

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✅ Solution

🔹 Step 1: Check Injection (One-One)

Take:

\[ x=2,\quad x=\frac12 \]

Then:

\[ f(2)=\frac{2}{5} \]

\[ f\left(\frac12\right)=\frac{\frac12}{\frac14+1} =\frac{\frac12}{\frac54} =\frac{2}{5} \]

Different inputs give same output.

❌ Not one-one.

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🔹 Step 2: Check Surjection (Onto)

Find range:

\[ f(x)=\frac{x}{x^2+1} \]

This function is bounded:

\[ -\frac12\leq f(x)\leq\frac12 \]

Maximum at:

\[ x=1,\quad f(1)=\frac12 \]

Minimum at:

\[ x=-1,\quad f(-1)=-\frac12 \]

So range:

\[ \left[-\frac12,\frac12\right] \]

But codomain is:

\[ \mathbb{R} \]

Values like:

\[ 2,\ -3 \]

are not attained.

❌ 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 reciprocal values to test injection
• Rational bounded functions often fail onto
• Use derivative or known range to test surjection