Domain or range? I am doing some questions on contraction mapping and the first part of the question is: 

I am not sure what the question is asking for and how to do it.
Any help is hugely appreciated! thank you

You want to find the range of $f(x)$ on the domain $[0,3]$; that is, the set $I=\{f(x)\,|\,x\in[0,3]\}$. Since the function is continuous on $[0,3]$, the set will be equal to
$$I=[\min_{x\in[0,3]} f(x),\max_{x\in[0,3]} f(x)].$$
To find the minimum and the maximum of the function, take a look at the sign of its derivative:
$$\frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{x^2-2x}{2x+1}\right)=\frac{2(x^2+x-1)}{(2x+1)^2}$$
We see that the sign is determined by the sign of $x^2+x-1$, which is negative on $\left[0,\frac{\sqrt{5}-1}{2}\right]$, zero at $\frac{\sqrt{5}-1}{2}$, and positive on $\left[\frac{\sqrt{5}-1}{2},3\right]$.
Hence $$\min_{x\in[0,3]} f(x)=f\left(\frac{\sqrt{5}-1}{2}\right)=\frac{-3+\sqrt{5}}{2}$$ and $$\max_{x\in[0,3]} f(x)=\max\left\{f(0),f(3)\right\}=f(3)=\frac{3}{7},$$ so $I=\left[\frac{-3+\sqrt{5}}{2},\frac{3}{7}\right]$.
