What are some examples of bounded functions?

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What are some examples of bounded functions?

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Answer 1 · 2015-10-21T01:09:11

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

Explanation:

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

$sin (x)$ $sin(x)$ , $cos (x)$ $cos(x)$ , $arctan (x) = {tan}^{- 1} (x)$ $arctan(x)=tan^{-1}(x)$ , $\frac{1}{1 + x^{2}}$ $1/(1+x^2)$ , and $\frac{1}{1 + e^{x}}$ $1/(1+e^(x))$ 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]}

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What are some examples of bounded functions?

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