Examples of Two One-One Functions Whose Sum is Not One-One
📺 Video Explanation
📝 Question
Give examples of two one-one functions:
\[ f_1,f_2:\mathbb{R}\to\mathbb{R} \]
such that:
\[ (f_1+f_2)(x)=f_1(x)+f_2(x) \]
is not one-one.
✅ Solution
Take:
\[ f_1(x)=x \]
and:
\[ f_2(x)=-x \]
🔹 Check \(f_1\)
Function:
\[ f_1(x)=x \]
is one-one because different inputs give different outputs.
✔ One-one.
🔹 Check \(f_2\)
Function:
\[ f_2(x)=-x \]
is also one-one.
✔ One-one.
🔹 Sum Function
Now:
\[ (f_1+f_2)(x)=x+(-x)=0 \]
So:
\[ (f_1+f_2)(x)=0 \quad \text{for all }x \]
This is a constant function.
Constant functions are not one-one.
❌ Not one-one.
🎯 Final Answer
Examples:
\[ \boxed{f_1(x)=x,\quad f_2(x)=-x} \]
Then:
\[ (f_1+f_2)(x)=0 \]
which is not one-one.
🚀 Exam Shortcut
- Take opposite one-one functions
- Their sum becomes constant
- Constant functions are never injective