Is Quotient of Two One-One Functions Always One-One?

📺 Video Explanation

📝 Question

Suppose:

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

are non-zero one-one functions.

Is:

\[ \left(\frac{f_1}{f_2}\right)(x)=\frac{f_1(x)}{f_2(x)} \]

necessarily one-one?

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✅ Solution

No, quotient need not be one-one.

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🔹 Counter Example

Take:

\[ f_1(x)=x,\quad f_2(x)=x^2 \]

on:

\[ \mathbb{R}\setminus\{0\} \]

Both are non-zero and one-one on suitable restricted domain.

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🔹 Quotient Function

Then:

\[ \left(\frac{f_1}{f_2}\right)(x)=\frac{x}{x^2}=\frac1x \]

This is actually one-one.

So choose better example:

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Take:

\[ f_1(x)=x,\quad f_2(x)=2x \]

Both are one-one.

Then:

\[ \left(\frac{f_1}{f_2}\right)(x)=\frac{x}{2x}=\frac12,\quad x\neq0 \]

This is constant.

Constant function is not one-one.

❌ Hence quotient need not be one-one.

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🎯 Final Answer

\[ \boxed{\text{No, } \frac{f_1}{f_2} \text{ need not be one-one}} \]

Example:

\[ f_1(x)=x,\quad f_2(x)=2x \]

gives:

\[ \frac{f_1(x)}{f_2(x)}=\frac12 \]

which is not injective.

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🚀 Exam Shortcut

• Take proportional one-one functions
• Quotient becomes constant
• Constant ⇒ not injective