MATH 1070: Mathematics of Change, part I

We now list certain properties of functions to which we will often refer.

A function \(f\) is said to be odd if \(f(-x) = -f(x)\) for all \(x\) in the domain of \(f\text{.}\) (It is unclear what this would mean if the domain contains \(x\) but not \(-x\text{.}\)) Similarly an even function \(f\) is one satisfying \(f(-x) = f(x)\text{.}\)

A function \(f\) is said to be increasing if \(f(x) \leq f(y)\) for all values of \(x\) and \(y\) in the domain of \(f\) such that \(x \lt y\text{.}\) Informally, the value of an increasing function gets bigger if the argument gets bigger. If you change the requirement that \(f(x) \leq f(y)\) to the strict inequality \(f(x) \lt f(y)\text{,}\) this defines the notion of strictly increasing. Decreasing and strictly decreasing functions are defined analogously but with one inequality reversed: \(f\) is decreasing if \(f(x) \geq f(y)\) for all \(x,y\) satisfying \(x \lt y\text{.}\)

We can also say when a function is increasing or decreasing on a part of the domain: \(f\) is increasing on the open interval \((a,b)\) if the above inequality holds for all \(x,y \in (a,b)\text{.}\) For any point \(c \in (a,b)\text{,}\) we then also say that \(f\) is increasing at \(c\text{.}\) In other words, to say \(f\) is increasing at a point \(c\) means there is some \(a \lt c \lt b\) such that \(f\) is increasing on the open interval \((a,b)\text{.}\)