Animated Solution for Mathematics - Functions: Let $f(x)=[x{]}^2-[x+3]-3,x\in \mathbb{R}$, where $[\cdot ]$ is the greatest integer function. Then

We are presented with the function $f(x)=[x{]}^2-[x+3]-3$. The presence of the greatest integer function, denoted by the square brackets $[\cdot ]$, requires us to utilize its fundamental properties to simplify the expression.

We rely on the identity: for any integer $k$, $[x+k]=[x]+k$. In our case, $k=3$, which is clearly an integer. Thus, we can rewrite $[x+3]$ as $[x]+3$.

Substituting this into our original function, we obtain:

Expanding the expression carefully to avoid sign errors, we get:

To make this more intuitive, let us introduce a dummy variable $n=[x]$. Since the output of the greatest integer function is always an integer, $n$ must be an integer. Our function now takes the familiar quadratic form:

We can factorize this quadratic expression as:

We are looking for the region where $f(x)<0$, which implies:

Using the wavy curve method, we identify the critical points at $n=-2$ and $n=3$. The inequality holds for the interval:

Since $n$ must be an integer, the possible values for $n$ are $\{-1,0,1,2\}$.

Now, we must map these values back to the domain of $x$. For each integer $n$, the condition $[x]=n$ corresponds to the interval $[n,n+1)$.

Applying this to our set of integers, we obtain the following intervals: $[-1,0)$, $[0,1)$, $[1,2)$, and $[2,3)$.

Taking the union of these intervals, we find the final solution: $x\in [-1,3)$

This is the region where our function is negative. By simplifying the expression, transforming it into a standard quadratic, and mapping the discrete integer results back to the continuous domain, we have successfully conquered the staircase.