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Checking injectivity | TIFR 2013 problem 36

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.

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