Check Function \(f(x)=x^3+1\) on \( \mathbb{R} \)

📺 Video Explanation

📝 Question

Check whether the function

\[ f:\mathbb{R}\to\mathbb{R},\quad f(x)=x^3+1 \]

is:

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

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✅ Solution

🔹 Step 1: Check Injection (One-One)

Assume:

\[ f(x_1)=f(x_2) \]

Then:

\[ x_1^3+1=x_2^3+1 \]

So:

\[ x_1^3=x_2^3 \]

Cube function is strictly increasing on real numbers.

Therefore:

\[ x_1=x_2 \]

✔ Function is one-one.

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🔹 Step 2: Check Surjection (Onto)

Let:

\[ y\in\mathbb{R} \]

Need:

\[ x^3+1=y \]

So:

\[ x^3=y-1 \]

\[ x=\sqrt[3]{y-1} \]

Since cube root of every real number exists:

\[ x\in\mathbb{R} \]

✔ Every real number has pre-image.

✔ Function is onto.

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🎯 Final Answer

\[ \boxed{\text{f is one-one and onto}} \]

So:

✔ Injection
✔ Surjection
✔ Bijection

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🚀 Exam Shortcut

• Cubic functions on real numbers are strictly increasing
• Use inverse to prove onto
• Strictly increasing + full range = bijection