In an exercise they display the question "A Function whose Domain is [-8,6] is graphed below" effectively showing a graph of the points x = -8 and x = 6. Isn't a graph whose Domain [-8,6] usually [x,y] as in x = -8 and y = 6?
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12$\begingroup$ Huh? I have no idea what you're trying to ask. $\endgroup$– user296602Aug 15, 2017 at 3:22
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$\begingroup$ I've made some changes to the question, I hope it's better now. $\endgroup$– Eli S.Aug 15, 2017 at 3:23
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2 Answers
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You must familiarize yourself with some common notation.
In the context of functions, $[a, b]$ is a closed interval containing all real numbers between $a$ and $b$, inclusive. This can be written as a compound inequality as $a \leq x \leq b$. Similarly $(a, b)$ is the open interval containing all real numbers $x$ strictly between $a$ and $b$. As an inequality, this is $a < x < b$.
The exercise in question speaks of a function whose domain is the interval $[-8, 6]$, that is, all $x$ for which $-8 \leq x \leq 6$. This is not a point.
When we speak of points, it will usually be clear in context. Domain talks about $x$-values (or whatever letter is used for the independent variable).
answered Aug 15, 2017 at 3:35
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5$\begingroup$ You might mention that in addition "$(x,y)$" can denote a point, but "$[x,y]$" does not denote something point-like. $\endgroup$ Aug 15, 2017 at 4:59
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$\begingroup$ nb a closed interval includes its endpoints inclusive, not exclusive (as in exclude). I've thus edited that part. $\endgroup$– AlexRAug 15, 2017 at 6:43
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$\begingroup$ I can't believe I missed that. Thank you, @AlexR $\endgroup$ Aug 15, 2017 at 6:44
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$[-8,6]$ is the set of all real numbers between $-8$ and $6$, inclusive.
$(-8,6)$ is the same set, only without $-8$ or $6$. It can also denote the point $x=-8, y=6$. OP is correct in that this is two different uses of notation.
An answer to this question would be $\sqrt{(x+8)(6-x)}$. If $x$ is outside the interval $[-8,6]$, then the term $(x+8)(6-x)$ will be negative, so outside the domain for the square root.
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$\begingroup$ +1. Just a remark: there's no need to be so contrived with the example: you could just take $\mathrm{id}: [-8,6]\to [-8, 6]$ to get a function like that. $\endgroup$ Aug 15, 2017 at 3:51
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$\begingroup$ @YoTengoUnLCD While that's true mathematically, this sounds like a precal homework question, and I think a precal teacher would count "id on the appropriate domain" as a wrong answer. $\endgroup$ Aug 15, 2017 at 5:13
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5$\begingroup$ @SilvioMayolo You're probably right on what you think the teacher might say, but I think that would be a terrible answer that should be eradicated, as it terribly confuses students, making them think that functions, and their domains are based on their "recipes". $\endgroup$ Aug 15, 2017 at 5:15