Example of a Function Which is Onto but Not One-One

📺 Video Explanation

📝 Question

Give an example of a function which is:

(ii) not one-one but onto.

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✅ Solution

Consider the function:

\[ f:\mathbb{R}\to [0,\infty) \]

defined by:

\[ f(x)=x^2 \]

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🔹 Check One-One

Take:

\[ x=2,\quad x=-2 \]

Then:

\[ f(2)=4,\quad f(-2)=4 \]

Different inputs give same output.

❌ Not one-one.

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🔹 Check Onto

Codomain:

\[ [0,\infty) \]

For any:

\[ y\geq0 \]

choose:

\[ x=\sqrt y \]

Then:

\[ f(x)=x^2=y \]

So every element in codomain has pre-image.

✔ Onto.

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🎯 Final Answer

An example is:

\[ \boxed{f(x)=x^2,\quad f:\mathbb{R}\to[0,\infty)} \]

This function is onto but not one-one.

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🚀 Exam Shortcut

• \(x^2\) repeats values for \(x\) and \(-x\)
• Choose codomain carefully for onto
• Standard example: square function