Check Function \(h(x)=x^2\) on \([-1,1]\)

📺 Video Explanation

📝 Question

Let:

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

Discuss whether the function:

\[ h:A\to A,\quad h(x)=x^2 \]

is one-one, onto, or bijective.

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✅ Solution

🔹 Check One-One (Injective)

Take:

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

Then:

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

Different inputs give same output.

❌ Not one-one.

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🔹 Check Onto (Surjective)

Since:

\[ x^2\geq0 \]

Range:

\[ [0,1] \]

But codomain:

\[ [-1,1] \]

Negative values are not attained.

Example:

\[ -\frac12 \]

has no pre-image.

❌ Not onto.

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🎯 Final Answer

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

So:

❌ Injection
❌ Surjection
❌ Bijection

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🚀 Exam Shortcut

• Square repeats positive/negative values
• Range becomes only non-negative
• So not injective and not onto