Check One-One / Onto

🎥 Video Explanation

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📝 Question

Let \( f:\mathbb{R}\setminus\{n\} \to \mathbb{R} \) be defined by

\[ f(x)=\frac{x-m}{x-n}, \quad m \ne n \]

• A. one-one onto
• B. one-one into
• C. many-one onto
• D. many-one into

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✅ Solution

🔹 Step 1: Check Injective

Assume \(f(x_1)=f(x_2)\):

\[ \frac{x_1-m}{x_1-n}=\frac{x_2-m}{x_2-n} \]

Cross multiply:

\[ (x_1-m)(x_2-n)=(x_2-m)(x_1-n) \]

Simplifying gives:

\[ (m-n)(x_1-x_2)=0 \]

Since \(m \ne n\), we get:

\[ x_1=x_2 \]

✔️ Function is one-one

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🔹 Step 2: Check Surjective

Let \(y=\frac{x-m}{x-n}\)

Solve for \(x\):

\[ y(x-n)=x-m \]

\[ yx-yn=x-m \]

\[ x(y-1)=yn-m \]

\[ x=\frac{yn-m}{y-1} \]

This is defined only when \(y \ne 1\).

So range: \[ \mathbb{R} \setminus \{1\} \]

Codomain is \(\mathbb{R}\), so 1 is missing.

❌ Not onto

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🔹 Final Answer

\[ \boxed{\text{Option B: one-one into}} \]