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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.