bsmart36

2022-08-11

Is a Square Bracket Used in Intervals of Increase/Decrease?

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

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

Cynthia George

Beginner2022-08-12Added 10 answers

Step 1

For $a,b\in \mathbb{R},a<b$, real intervals are defined as follows:

$(a,b):=\{x\in \mathbb{R}\mid a<x<b\}$

$(a,b]:=\{x\in \mathbb{R}\mid a<x\le b\}$

$[a,b):=\{x\in \mathbb{R}\mid a\le x<b\}$

$[a,b]:=\{x\in \mathbb{R}\mid a\le x\le b\}$

Step 2

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], $(a,\mathrm{\infty})$, $[a,\mathrm{\infty})$ with $a,b\in \mathbb{R}$, $a<b\le 0$ (these are all intervals an which f decreases).

For $a,b\in \mathbb{R},a<b$, real intervals are defined as follows:

$(a,b):=\{x\in \mathbb{R}\mid a<x<b\}$

$(a,b]:=\{x\in \mathbb{R}\mid a<x\le b\}$

$[a,b):=\{x\in \mathbb{R}\mid a\le x<b\}$

$[a,b]:=\{x\in \mathbb{R}\mid a\le x\le b\}$

Step 2

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], $(a,\mathrm{\infty})$, $[a,\mathrm{\infty})$ with $a,b\in \mathbb{R}$, $a<b\le 0$ (these are all intervals an which f decreases).

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