We want to get a good grasp of $f(x)$. One way I would recommend is to draw the graph $y=f(x)$. (If necessary, you might have some software do the drawing, but don't necessarily trust the result.) Regrettably, I will have to do things without a picture.
By completing the square, we see that $f(x)=(x-2)^2-4+3=(x-2)^2-1$. So the curve $y=f(x)$ is a parabola. Now we can trace out $f(x)$ as $x$ travels from $1$ to $4$.
At $x=1$ (which is not in the interval $(1,4]$), we have $f(x)=0$. Then as $x$ travels from $1$ to $2$, $f(x)$ decreases, until it reaches $-1$ at $x=2$. So the vertex of the parabola is at $(2,-1)$. Then, as $x$ increases from $2$ to $4$, $(x-2)^2-1$ increases from $-1$ to $3$.
So all values from $-1$ to $3$, inclusive, are taken on by $f(x)$, as $x$ travels over the interval $(1,4]$. The answer is therefore $[-1,3]$.