Product of Two One-One Functions Need Not Be One-One

📺 Video Explanation

📝 Question

Show that if:

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

are one-one, then:

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

need not be one-one.

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✅ Solution

Take:

\[ f_1(x)=x \]

and:

\[ f_2(x)=\frac{1}{x},\quad x\neq0 \]

(Or restrict domain excluding 0.)

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🔹 Check \(f_1\)

Function:

\[ f_1(x)=x \]

is one-one.

✔ One-one.

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🔹 Check \(f_2\)

Function:

\[ f_2(x)=\frac1x \]

is also one-one on:

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

✔ One-one.

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🔹 Product Function

Now:

\[ (f_1\times f_2)(x)=x\cdot\frac1x=1 \]

for all:

\[ x\neq0 \]

This is constant.

Constant function is not one-one.

❌ Hence product need not be one-one.

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🎯 Final Answer

Example:

\[ \boxed{f_1(x)=x,\quad f_2(x)=\frac1x} \]

Then:

\[ (f_1\times f_2)(x)=1 \]

which is not one-one.

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🚀 Exam Shortcut

• Choose inverse-type functions
• Product becomes constant
• Constant ⇒ not injective