What Are the Nuances of Mathematical Symbols and Their Applications?

[jaysquestions]

None of these questions are homework problems, they're just confusing things that homework has caused

1. In math, what is the difference between a bracket and a parenthesis? Also what about the right symbols for ordered pairs , is that strictly these [ ] or these { }

2. If you have lots of brackets (just to be clear, I mean these ( ) )... does the number of left hand brackets always equal the right hand brackets? For example,

$f(x) = (3(x^2 - 3) / x(x^3+6))$ I realize it could be written a lot different, but I am just focused on the brackets.

..so when you are checking over a long problem, you can count the left brackets and make sure they = right brackets. Is that valid? (in English one right bracket can close multiple left brackets, which has never made sense to me. So math is different?)

3. What is he difference between "broadened" and "steepened" wrt graphs?

4. Are corners of a function defined and how do I know they are defined? (or does it all depend on the function?

5. What does the textbook mean when he says "use analytical method to determine..." Is he just saying make a graph and look at it?

6. Conic sections: Why are hyperbolas represented with two lines , taken from two cones, when circles and ellipses are represented using just one cone?

thank you for any answers

 

[Algr]

... does the number of left hand brackets always equal the right hand brackets?

Yes. If a closing parenthesis is missing, there is no reason to assume it belonged at the end. It's a mistake.

n English one right bracket can close multiple left brackets, which has never made sense to me. So math is different?

I've never heard of this. Brackets and parentheses must always be used in pairs. The only exceptions are smilies and leetspeak.

 

[jedishrfu]

LISP programming used to use the convention of a square bracket closing the program statement. It would basically close all remaining open parentheses.

Also there's a general convention in math to call "( )" parentheses and "[ ]" square brackets and "{ }" curly braces with some folks dropping the square and curly adjectives.

Using an analytical method means to use algebra rules, trigonometric, boolean, ... identities and Calculus rules to reduce an expression into a simpler form or to use numerical methods to get an answer. This is in opposition to graphing the function and determining things visually by inspection.

 

[HallsofIvy]

None of these questions are homework problems, they're just confusing things that homework has caused

1. In math, what is the difference between a bracket and a parenthesis? Also what about the right symbols for ordered pairs , is that strictly these [ ] or these { }

In general there is no difference. There may be special cases. For example, you ask about "ordered pairs" being either [ ] or { }. I have never seen either of those. I have always seen (a, b) for ordered pairs.

2. If you have lots of brackets (just to be clear, I mean these ( ) )... does the number of left hand brackets always equal the right hand brackets? For example,

$f(x) = (3(x^2 - 3) / x(x^3+6))$ I realize it could be written a lot different, but I am just focused on the brackets.

Yes, "brackets" ("braces", "parentheses") always separate a given section of a formula from the rest. You should never have a left bracket without a corresponding right and vice versa.

..so when you are checking over a long problem, you can count the left brackets and make sure they = right brackets. Is that valid? (in English one right bracket can close multiple left brackets, which has never made sense to me. So math is different?)

I have never heard of such an "English rule". Where did you learn that? Is it, for example, in Strunk's "The Elements of Style"?

3. What is he difference between "broadened" and "steepened" wrt graphs?

Pretty much just what it says. If you have a function, y= f(x), such that f(1)= 3, f(2)= 4, and f(4)= 7, so that the graph contains the points (1, 3), (2, 4), and (4, 7), then the graph of y=f(2x) satisfies f(2)= 3 and f(4)= 4 so contains the points (2, 3) and (4, 4). This new graph takes twice as long (2 to 4) to go from height 3 to height 4 a before (1 to 2) so has "broadened. On the other hand, the graph of the function y= 2f(x) contains the points (1, 6) and (2, 8). In going from x= 1 to x= 2 horizontally, it has gone, vertically, from 6 to 8 so is steeper.

4. Are corners of a function defined and how do I know they are defined? (or does it all depend on the function?

I have no idea what you mean here. A "function" is not a geometric object and so does NOT have "corners". Where did you see "corners of a function"? How were they defined where you saw the reference?

5. What does the textbook mean when he says "use analytical method to determine..." Is he just saying make a graph and look at it?

No, an "analytical method" (just "analytical method" without an article would be bad English!) is one that involves limits and calculations, not just looking at a graph. It might mean considering continuity and/or using limits to determine what asymptotes there are.

6. Conic sections: Why are hyperbolas represented with two lines , taken from two cones, when circles and ellipses are represented using just one cone?

Because that is the way they are created. circles, ellipses (and parabolas) have only one connected component because they are created by slicing a cone (grammatical point: strictly speaking a "cone" always has two parts, not just one. It is incorrect to say "two cones" or "just one cone". You are rather, referring to one or two components of a cone.) with a plane. In the case of a circle so that the plane is perpendicular to the axis of the cone, and only intersects one component, in the case of an ellipse so that it is between perpendicular to the axis and parallel to one side, so that it still slices through one component of the cone, in the case of a parabola parallel to the side of the cone so goes through one side of one component and is parallel to the other, and, finally, a hyperbola that crosses through both components.

thank you for any answers

 

[SteamKing]

None of these questions are homework problems, they're just confusing things that homework has caused

3. What is he difference between "broadened" and "steepened" wrt graphs?

What do the words "broad" and "steep" mean to you?

4. Are corners of a function defined and how do I know they are defined? (or does it all depend on the function?

Corners of what function? You'll have to be more specific here.

5. What does the textbook mean when he says "use analytical method to determine..." Is he just saying make a graph and look at it?

If that were the case, then the textbook would say, "graph [whatever] to determine ..."

Analytical methods usually involve doing calculations of some sort.

6. Conic sections: Why are hyperbolas represented with two lines , taken from two cones, when circles and ellipses are represented using just one cone?

Look up how conic sections are defined by a plane which intersects the cone.

Some of these questions you could have answered yourself by doing a little research.

The web is a wonderful tool, but you've got to put forth a little effort to get it to yield its information.

Try using a search engine to answer some of these questions.

 

[Hector Mata]

Hyperbolas come from double cones:

In this image, that would be #3. This is what https://www.physicsforums.com/members/hallsofivy.331/ meant when he said:

strictly speaking a "cone" always has two parts, not just one. It is incorrect to say "two cones" or "just one cone". You are rather, referring to one or two components of a cone.