Example of a Function Which is Neither One-One Nor Onto

📺 Video Explanation

📝 Question

Give an example of a function which is:

(iii) neither one-one nor onto.

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✅ Solution

Consider the function:

\[ f:\mathbb{R}\to\mathbb{R} \]

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:

\[ \mathbb{R} \]

But:

\[ f(x)=x^2\geq0 \]

So negative real numbers are never obtained.

Example:

\[ -1 \] has no pre-image.

❌ Not onto.

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🎯 Final Answer

An example is:

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

This function is neither one-one nor onto.

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🚀 Exam Shortcut

• Square function repeats values → not one-one
• Negative outputs missing → not onto
• Very common exam example