I have to consider the functions $f$ and $g$ defined on $\mathbb{R}$ by $$f(x)=(2x-1)^2 \quad \text{ and }\quad g(x)=\sup_{x \leq t \leq x+1}f(t)$$
The goal is to compute explicitely $g(x)$. Unfortunately, I have never done such an exercicse before, and I do not know how to attack such a problem. Any help is therefore welcome! Thank you in advance!
My advice would be to pick a particular $x$-value, and then compute $g(x)$. You will start to notice a pattern after you do a few of these hopefully.
For example, for $x = 4$, we are interested in $g(4) = \sup_{4 \leq t \leq 5} f(t)$. Since $f(x)$ is continuous, the function actually attains a max (and min) on each closed interval $[x,x+1]$ by the extreme value theorem. If you look at a graph for your function, you'll notice that on $[4,5]$, the function $f$ is increasing so that the $\sup$ (max) comes at the right endpoint of the interval, $f(5) = 81$. This means that $g(4) = 81$.
Do a bunch of these until you get the hang of what's happening, and pay close attention to the graph on the closed interval you're interested in. Then you can think about generalizing and giving an explicit formula for $g(x)$.