Check Function \(f(x)=x-5\) on \( \mathbb{Z} \)

📺 Video Explanation

📝 Question

Check whether the function

\[ f:\mathbb{Z}\to\mathbb{Z},\quad f(x)=x-5 \]

is:

• injection (one-one)
• surjection (onto)
• bijection

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✅ Solution

🔹 Step 1: Check Injection (One-One)

Assume:

\[ f(x_1)=f(x_2) \]

Then:

\[ x_1-5=x_2-5 \]

So:

\[ x_1=x_2 \]

✔ Hence, function is one-one.

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🔹 Step 2: Check Surjection (Onto)

Let:

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

We need:

\[ f(x)=y \]

That is:

\[ x-5=y \]

So:

\[ x=y+5 \]

Since \(y+5\in\mathbb{Z}\), every integer has a pre-image.

✔ Hence, function is onto.

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🎯 Final Answer

\[ \boxed{\text{f is one-one and onto}} \]

So:

✔ Injection
✔ Surjection
✔ Bijection

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🚀 Exam Shortcut

• Linear functions like \(x+c\) on integers are bijections
• Equal outputs give equal inputs
• Reverse formula helps prove onto