Check Function \(f(x)=5x^3+4\) on \( \mathbb{R} \)

📺 Video Explanation

📝 Question

Check whether the function

\[ f:\mathbb{R}\to\mathbb{R},\quad f(x)=5x^3+4 \]

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:

\[ 5x_1^3+4=5x_2^3+4 \]

So:

\[ x_1^3=x_2^3 \]

Thus:

\[ x_1=x_2 \]

✔ Function is one-one.

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🔹 Step 2: Check Surjection (Onto)

Let:

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

Need:

\[ 5x^3+4=y \]

So:

\[ 5x^3=y-4 \]

\[ x^3=\frac{y-4}{5} \]

\[ x=\sqrt[3]{\frac{y-4}{5}} \]

Since cube root exists for every real number:

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

✔ 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

• Odd degree polynomials with positive leading coefficient are increasing
• Use inverse form for onto proof
• Strictly increasing + all real outputs = bijection