Proof that the function $f(x)=x(\frac{3}{2}+\lfloor -x^2 \rfloor)$ is injective

Question

I've been stuck on this one for a while, what I've basically tried is showing that:
$$ f(x) = f(y)\ \ \Longrightarrow\ \ x = y$$
Can anyone help?

Answer 1

We can show $f$ is injective by the $3$ observations below:

• As $x>1$ increases, $\lfloor-x^2\rfloor\leq-2$ is decreasing, thus $f(x)>-1/2$ is strictly decreasing as $x>1$ increases.

• As $x<-1$ decreases, $\lfloor-x^2\rfloor\leq-2$ is decreasing, thus $f(x)>1/2$ is strictly increasing as $x<-1$ decreases.

• For $|x|\leq1$, we have $\lfloor-x^2\rfloor=-1$, so $f(x)=x/2\in[-1/2,1/2]$.

Here is a Desmos plot to help understand this argument.