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

📺 Video Explanation

📝 Question

Check whether the function

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

is:

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

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✅ Solution

🔹 Step 1: Simplify Function

Using identity:

\[ \sin^2x+\cos^2x=1 \]

So:

\[ f(x)=1 \quad \text{for all } x\in\mathbb{R} \]

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🔹 Step 2: Check Injection (One-One)

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

But:

\[ f(0)=1,\quad f(1)=1 \]

Different inputs give same output.

❌ Not one-one.

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🔹 Step 3: Check Surjection (Onto)

Codomain:

\[ \mathbb{R} \]

Range:

\[ \{1\} \]

Only value 1 is attained.

So numbers like:

\[ 0,\ 2,\ -1 \]

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

• Use trigonometric identity first
• Constant functions are never one-one
• Constant function is onto only if codomain has one element