Examples of Two Surjective Functions Whose Sum is Not Surjective

📺 Video Explanation

📝 Question

Give examples of two surjective functions:

\[ f_1,f_2:\mathbb{Z}\to\mathbb{Z} \]

such that:

\[ (f_1+f_2)(x)=f_1(x)+f_2(x) \]

is not surjective.

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✅ Solution

Take:

\[ f_1(x)=x \]

and:

\[ f_2(x)=x \]

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🔹 Check \(f_1\)

For any:

\[ y\in\mathbb{Z} \]

choose:

\[ x=y \]

Then:

\[ f_1(x)=y \]

✔ Surjective.

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🔹 Check \(f_2\)

Similarly:

\[ f_2(x)=x \]

is surjective.

✔ Surjective.

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🔹 Sum Function

Now:

\[ (f_1+f_2)(x)=x+x=2x \]

Range:

\[ \{ \text{even integers} \} \]

Odd integers are not attained.

Example:

\[ 1,3,5 \]

have no pre-image.

❌ Not surjective.

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🎯 Final Answer

Examples:

\[ \boxed{f_1(x)=x,\quad f_2(x)=x} \]

Then:

\[ (f_1+f_2)(x)=2x \]

which is not surjective on \(\mathbb{Z}\).

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🚀 Exam Shortcut

• Take simple onto functions
• Make sum restricted (like even numbers)
• Missing outputs means not onto