Let's solve a problem based on checking injectivity from TIFR 2013 problem 36. Try it yourself first, then read the solution here.
Question:
True/False?
The function \(f:\mathbb{Z} \to \mathbb{R} \) defined by \(f(n)=n^3-3n\) is injective.
Hint:
Check for small values!
Discussion:
Surely, checking for small values will give you that f is not injective.
For example, let us look at \(f(0)=0\), \(f(1)=-2\), \(f(-1)=2\), \(f(2)=2\).
So \(f(-1) = f(2)\).
Therefore, \(f\) is not injective.