Check Function \(g(x)=|x|\) on \([-1,1]\)

📺 Video Explanation

📝 Question

Let:

\[ A=[-1,1] \]

Discuss whether the function:

\[ g:A\to A,\quad g(x)=|x| \]

is one-one, onto, or bijective.

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✅ Solution

🔹 Check One-One (Injective)

Take:

\[ x=1,\quad x=-1 \]

Then:

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

Different inputs give same output.

❌ Not one-one.

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🔹 Check Onto (Surjective)

Since:

\[ |x|\geq0 \]

Range of function:

\[ [0,1] \]

But codomain is:

\[ [-1,1] \]

Negative values like:

\[ -\frac12,\ -1 \]

are not attained.

❌ Not onto.

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🎯 Final Answer

\[ \boxed{\text{g is neither one-one nor onto}} \]

So:

❌ Injection
❌ Surjection
❌ Bijection

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🚀 Exam Shortcut

• Modulus gives same value for ±x → not injective
• Range becomes non-negative only
• Compare interval carefully