Pondering of the fundamentals of graphing functions

Question

I was doing my calculus homework when I realized I didn't fully understand how graphing functions works. My questions is: is $f(x)$ a curve or is $f$ a curve? My thinking is that $f(x)$ is a specific but unknown ordinate (not coordinate) because the input x (or abscissa) is a specific but unknown value. That is why we name the vertical axis on a coordinate plane $f(x)$; we are plotting infinitely many f(x)s along with their corresponding x's to generate the curve f. My second question is that if I have a line segment of the curve $y=3$, I know it is made up of infinitely many points. Now if I cut off a line segment that is bigger than my first line segment, this new line segment is still made up of infinitely many points. But because the line segment is longer than before, do we have a case of one infinity being greater than the other?