This is one of the first examples of manifolds, from John M. Lee - Introduction to Smooth Manifolds.
I'm having trouble to understand two things, which I'm gonna show in the simple case of $n=2$. In this case, I'll write $x,y,z$ instead $x^1, x^2, x^3$.
Now we have $f:\mathbb{B}^2\to\mathbb{R}$ such that $f(x,y,z) = \sqrt{1-x^2-y^2-z^2}$, and we have that $U_1^+\cap\mathbb{S}^2$ is "right side" of the sphere. My problems starts when he says this is the graph of the function $$x = f(y,z).$$
The first problem is: $f$ is defined in three variables, so it makes no sense to omit some of the variables.
The second problem is: how this equation defines a function? It's no clear what should be the input (domain), what should be the output (image) and what is the function itself.
Thanks for the help.
You are counting incorrectly. $\mathbb{B}^2$ is two dimensional, so you will get $f=f(x,y)$, say (or $f(y,z)$ or $f(x,z)$, according to your choice). Assuming you are using the first choice you'll write the sphere as $z=f(x,y)$ with $(x,y) $ chosen from $\mathbb{B}^2$, where $\sqrt{1-x^2-y^2}$ is well defined.