For example, the I.O.I of y=x^2 is (0,infinite), with the round brackets meaning that the value is excluded. Are there any scenarios where a square bracket would be used when stating the intervals of increase/ decrease for a function? If it narrows it down, the only functions I deal with are: linear, exponential, quadratic, root, reciprocal, sinusoidal, and absolute
1 Answer
For $a,b\in\mathbb R,a<b$, real intervals are defined as follows: $$(a,b):=\{x\in\mathbb R\mid a<x<b\}$$ $$(a,b]:=\{x\in\mathbb R\mid a<x\leq b\}$$ $$[a,b):=\{x\in\mathbb R\mid a\leq x<b\}$$ $$[a,b]:=\{x\in\mathbb R\mid a\leq x\leq b\}$$ Each function is defined on domain. If the domain is a subset of $\mathbb R$ that contains intervals, you can ask which behavior the function has on these intervals.
For example, $f(x)=x^2,x\in\mathbb R$
- is increasing on any interval $(a,b)$, $(a,b]$, $[a,b)$, $[a,b]$, $(a,\infty)$, $[a,\infty)$ with $a,b\in\mathbb R$, $0\leq a<b$ (these are all intervals on which $f$ increases) and
- decreasing on any interval $(a,b)$, $(a,b]$, $[a,b)$, $[a,b]$, $(\infty,b)$, $(\infty,b]$ with $a,b\in\mathbb R$, $a<b\leq 0$ (these are all intervals an which $f$ decreases).
