Check Function \(f(x)=x^3-x\) on \( \mathbb{R} \)

📺 Video Explanation

📝 Question

Check whether the function

\[ f:\mathbb{R}\to\mathbb{R},\quad f(x)=x^3-x \]

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:

\[ x=-1,\quad x=0,\quad x=1 \]

Then:

\[ f(-1)=(-1)^3-(-1)=0 \]

\[ f(0)=0 \]

\[ f(1)=1-1=0 \]

Different inputs give same output.

❌ Not one-one.

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🔹 Step 2: Check Surjection (Onto)

Since:

\[ \lim_{x\to\infty}(x^3-x)=\infty \]

and:

\[ \lim_{x\to-\infty}(x^3-x)=-\infty \]

Function is continuous on \(\mathbb{R}\).

By Intermediate Value Theorem, it takes every real value.

✔ Onto.

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🎯 Final Answer

\[ \boxed{\text{f is onto but not one-one}} \]

So:

❌ Injection
✔ Surjection
❌ Bijection

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🚀 Exam Shortcut

• Try small values for injection test
• Odd degree continuous polynomials cover all reals
• So cubic-type functions are onto