If there is always │f(x)│≤M for all x values belonging to an interval I, where m is a constant that has nothing to do with x, then we say that f(x) is bounded in the interval I, otherwise it is called unbounded.
For example, y=arctanx is bounded on the whole real number field.
You can visually find two boundaries, one is y=π/2, and the other is y=-π/2. All the function values are outside this range.
If a function has a minimum and a maximum, it must be bounded.
The maximum and minimum values are boundaries.
The most vivid image of unbounded function is y=tanx. When x approaches π/2, the function value approaches infinity.