Show \(f(x)=x^2\) is Neither One-One Nor Onto

📺 Video Explanation

📝 Question

Let:

\[ A=\{-1,0,1\} \]

and function:

\[ f=\{(x,x^2):x\in A\} \]

Show that:

\[ f:A\to A \]

is neither one-one nor onto.

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✅ Solution

🔹 Step 1: Find Function Values

For:

• \(f(-1)=(-1)^2=1\)
• \(f(0)=0^2=0\)
• \(f(1)=1^2=1\)

So:

\[ f=\{(-1,1),(0,0),(1,1)\} \]

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🔹 Step 2: Check One-One

A function is one-one if different inputs give different outputs.

Here:

\[ f(-1)=1,\quad f(1)=1 \]

But:

\[ -1\neq1 \]

Different inputs give same output.

❌ Not one-one.

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🔹 Step 3: Check Onto

Codomain:

\[ A=\{-1,0,1\} \]

Range:

\[ \{0,1\} \]

Element:

\[ -1 \]

is not in range.

❌ Not onto.

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🎯 Final Answer

\[ \boxed{f:A\to A \text{ is neither one-one nor onto}} \]

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🚀 Exam Shortcut

• Square function often repeats values
• List all mappings first
• Compare range with codomain