Question

What are some examples of bounded functions?

Answer 1

`sin(x)`, `cos(x)`, `arctan(x)=tan^{-1}(x)`, `1/(1+x^2)`, and `1/(1+e^(x))` are all commonly used examples of bounded functions.

Explanation:

A function `f(x)` is bounded if there are numbers `m` and `M` such that `m leq f(x) leq M` for all `x`. In other words, there are horizontal lines the graph of `y=f(x)` never gets above or below.

`sin(x)`, `cos(x)`, `arctan(x)=tan^{-1}(x)`, `1/(1+x^2)`, and `1/(1+e^(x))` are all commonly used examples of bounded functions (as well as being defined for all `x in RR`). There are plenty more examples that can be created.

The graph of `1/(1+e^(x))` is interesting because it has two distinct horizontal asymptotes (`arctan(x)` does too). The graph of `1/(1+e^(x))` is shown below.

graph{1/(1+e^(x)) [-5, 5, -2.5, 2.5]}