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

📺 Video Explanation

📝 Question

Check whether the function

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

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 give different outputs.

Take:

\[ x_1=0,\quad x_2=2\pi \]

Then:

\[ \sin0=0,\quad \sin2\pi=0 \]

But:

\[ 0\neq2\pi \]

❌ Not one-one.

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🔹 Step 2: Check Surjection (Onto)

Range of sine function is:

\[ [-1,1] \]

But codomain is:

\[ \mathbb{R} \]

Values like:

\[ 2,\ -3 \]

are never 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

• Sine is periodic → not one-one
• Range is only \([-1,1]\)
• Not onto over all real numbers