Question

Why f(x) and g(x) do not cut the x-axis at the same number of points. Why h(x) does not cut the x-axis. Why do all cut the y-axis at one point only.

Answer 1

You probably have a graph of the three relations/functions in front of you, but we do not see it.

However, if f(x), g(x), and h(x) are functions, then the following statements are true:
1. Any defined function can intersect the x-axis any number of times, depending on the definition of the function. For example, the sine function cuts the x-axis infinite number of times.
2. By the definition of a function, any vertical line through any point on the domain of the function can intersect the function at exactly one point.
That is why the all the functions cut the y-axis at (at most) one point.