Talk:Calculus/Functions

Kids Section
The final section about "Algebraic manipulation" is really strange, i think the author must assume that the reader of that advanced topic "Calculus" already have this very basic knowledge introduced in that section. i suggest omitting it to make the book more concise and aiming

Would it be worth noting in the Graphing Functions section that graphs are usually of the form y=mx+b, rather than the example which used x=my+b? i.e.:

Functions may be graphed by finding the value of f for various y and plotting the points (y, f(y)) in a Cartesian plane...

2(x+2)/2 = x+2. That's not incorrect. Maybe you meant 2x+2/2 ≠ 2 (x+2)/2 = x+2?

Is a function considered a variable or a rule? f(z) doesn't vary with x; f(x) doesn't vary with z. f(x) may be a variable, but f by itself is a rule or a mapping (or a "machine", as textbooks say). y is a dependent variable because it is understood to vary with x -- y(x) is never used.

What's the plus function doing? + is more common; plus(2,2) is almost never used except in Lisp or something. At least call them add, subtract, divide, multiply instead. Using plus(plus(x,y),z) is not the best way to show grouping because (x+y)+z is more common. Even you use 1/(x+2) at the bottom (instead of divide(1,plus(x,2))). --Geoffrey 15:37 23 Jul 2003 (UTC)

Points
You have some good points (and one misreading):

I did not write "2(x+2)/2 = x + 2", but rather "2(x+2)/2 = (x+2)/2", and the latter is incorrect.

As for the issue of a function being a variable or rule, I do not make a distinction in these informal pages between the 'rule' (i.e., function), or the result of the rule (i.e., the value of the function). The value of the function is indeed a variable. Making a big deal about distinguishing the two (very different) concepts may only confuse the matter, but an astute reader make take things 'too' literally and notice inconsistancies as you did. The best way of teaching functions is a matter of opinion. Arguably, one should teach functions in a nonstandard way to expose readers to different perspectives or methods, but also arguably, the standard way of teaching functions is standard because of its superiority.

The use of functions plus(x, y) and times(x, y) is to emphasize that the reader has already been doing functions for a long time, and to give the reader something firm to latch onto to relate abstract concepts of arbitrary functions to specific, familar functions. In particular, the composition of functions is confusing, but most people can compose the 'plus' and 'times' functions with ease. Of course, writing out 'plus' and 'times' is somewhat unnecessary.

At User Talk:Eric119 is some more discussion about Calculus/Functions.

If you see anything that needs modification, just go ahead and modify it!

-- IntMan

A graph is a locus of points in the Cartesian plane

I think that the majority of people who can understand the phrase "locus of points in the Cartesian plane" probably already know what a graph is. I think the fact that the majority of us have done maths to a university or higher level is causing us to use language that may be over-complex for our target audience.

--Imran 12:50 27 Jul 2003 (UTC)


 * Good point. Also, thanks for the exercises. -- IntMan

Hey, why take out my Example section in basic Calculus??


 * Reverted to get rid of the lines, but reverted too far. Sorry. 129.94.6.30 22:17, 11 Feb 2004 (UTC) (User:Dysprosia, logged out)

Ok, thanks :) The lines aren't formatting ppl want? --LWM
 * Horizontal rules aren't pretty, and should be used sparingly. Nice exercise, by the way... Dysprosia 20:49, 12 Feb 2004 (UTC)

- Could someone please clarify the following?:


 * If the language needs to be more clear here, it translates to:


 * Suppose there is a function $$f(x)=\sqrt{x}$$ whose domain is $$\{x|x>0|x\in\mathbb{R}\}$$.
 * }
 * The $$\sqrt{x}$$ basically asks what is the value of $$a$$ so that $$a^2=x$$, the domain encompasses values of $$x$$ such as $$0,2,\frac{1}{2},e,\pi,\text{etc.}$$ but not any negative numbers. As is written currently, this is clear. Musical Inquisit (discuss • contribs) 09:12, 19 February 2021 (UTC)
 * }
 * The $$\sqrt{x}$$ basically asks what is the value of $$a$$ so that $$a^2=x$$, the domain encompasses values of $$x$$ such as $$0,2,\frac{1}{2},e,\pi,\text{etc.}$$ but not any negative numbers. As is written currently, this is clear. Musical Inquisit (discuss • contribs) 09:12, 19 February 2021 (UTC)

- I had something written, but I retract the question. It was a really dumb question. I really did need to brush up my calculus. GEez

don't know meaning of symbols: I took algebra 2 not too long ago and i have no idea what some of these symbols mean. for example the last function in the table of examples that ends in "Takes an input and uses it as boundary values for an integration." the thing that sort of looks like a sideways 8, and the exact meaning of these [] when in this form: (a,b];[a,b). Any chance of adding the meaning of these? or at least tell me a place where i can find them?--V2os 04:04, 31 July 2005 (UTC)


 * I think the last function on the table will be explained later. The sideways 8 stands for infinity. (a,b] is interval notation (first hit on google: http://id.mind.net/~zona/mmts/miscellaneousMath/intervalNotation/intervalNotation.html). And for the Set Notation the funny R means real numbers, and the funny E means the thing on the left is an element of the set on the right (x is an element of the set of real numbers). (disclaimer: I didn't know most of this before today ;) 4.228.240.29

--

Hrmm... but what happens with:

When x is -3? Isn't the function undefined at that point?
 * Yes. Musical Inquisit (discuss • contribs) 09:12, 19 February 2021 (UTC)

-- "A function f(x) has an inverse function if and only if f(x) is one-to-one." Is this true? It has to be onto function as well, isn't it?
 * Imagine the following function $$y=f(x)=2x^{3}$$, where $$x\in\mathbb{Z}$$ and $$y\in\mathbb{R}$$. This function is obviously not surjective since $$y=\frac{1}{2}$$ cannot be found for any $$x\in\mathbb{Z}$$. However, this function is definitely invertible. Take any points $$(x,y)$$ and map them so that the points read $$(y,x)$$. This is given by $$f^{-1}(x)=\sqrt[3]{\frac{x}{2}}$$, with $$x\in\mathbb{R}$$ and $$y\in\mathbb{Z}$$. Hopefully, I made clear how a function does not also need to be onto. Musical Inquisit (discuss • contribs) 09:12, 19 February 2021 (UTC)

?
it seemse that wiki may have some problems... Look at the section on function manipulation. The formatting seems to have gone all wonky, and I have no idea why. THere is one part where it randomly substitutes a formula from the top of the page, instead of what is written there. If anyone has thoughts on how to fix, please do!

Problem sets
If this is supposed to be a textbook, where are the problem sets? It can be difficult to just read this and remember it all. I have half a mind to write a problem set. In fact, here's a few problems to get started (I'll leave the finishin' of it to someone more qualified... unless I get fed up):

Find the inverse of the following functions:


 * 1. $$f(x) = 3x^{2} + 4x$$
 * 2. $$f(x) = \frac{1+x}{2+x}$$

Evaluate:


 * 3.
 * $$(f\circ g)(x)$$ where
 * $$f(x) = \sqrt{x}$$ and
 * $$g(x) = 2x^{2}+3$$

216.215.128.84 (talk) 06:57, 15 December 2007 (UTC)


 * In this textbook, all exercises are part of a separate page. This has its own advantages and disadvantages, but that is the style this wikibooks went for. Musical Inquisit (discuss • contribs) 09:13, 19 February 2021 (UTC)

Length
Anyone else think this page is too long?Tiled (talk) 21:22, 17 November 2009 (UTC)

Translation left/right
Reading this, I got kind of a headache:

Horizontal translation by $$a$$ units left : $$f(x + a)$$

Horizontal translation by $$a$$ units right: $$f(x - a)$$

Is it possible that left and right have been interchanged? Please help me out ?

Andre anckaert (talk) 15:41, 15 January 2010 (UTC)
 * It's a little tricky to get your head around. Take f(x) = x2. Now, translate it left (you'll see): f(x + 1). What does x have to equal for y to equal zero (for the function to return zero)? It has to be -1, meaning, when y = 0, x must be -1, so the function was translated left. That's the best way I can think to explain it, hope it helped if you were still confused. DrSturm (talk) 16:38, 26 May 2010 (UTC)

important functions - quadratic
I am no mathematician and so will not consider editing this myself but the text as it is at present reads:

" A polynomial of the second degree. Its graph is a parabola, unless a = 0. (Don't worry if you don't know what this is.) "

I suspect the parenthesis text is reassuring the reader that knowledge of the parabola is not important, rather than not understanding when a=0 so it would be better written as:

"A polynomial of the second degree. Its graph is a parabola, (Don't worry if you don't know what a parabola is.) unless a = 0."

--JohnH99 (talk) 19:37, 18 April 2010 (UTC)


 * Your edit would be fine! Trust yourself and be bold, there are very few mathematicians around here.  Nothing invalidates the mathematics you know and every editor here (or on any wikimedia project) is on equal footing. Holding a mathematics degree gains you no additional credit or influence.  People are supposed to be judged solely on the quality of their edits, and I have found many non-mathematicians do a really great job editing math articles.  It seems they still remember how to explain the confusing parts, where if you've been studying it for 20 years, you forget which parts were confusing.   And if you make a mistake then (hopefully) someone will fix and and be civil in doing so. Thenub314 (talk) 20:09, 18 April 2010 (UTC)

Bijective and surjective
41.222.2.22 correctly points out these ideas are notable absent form this page and should probably be added. Thenub314 (talk) 16:44, 5 June 2010 (UTC)


 * I added these concepts, in the Modern Understanding of Functions section, so no need to worry. Musical Inquisit (discuss • contribs) 15:50, 31 March 2020 (UTC)

A function in everyday life has X, Y, and Z, not just X and Y
This is a function of two real variables (a location is described by two values - an x and a y ) which results in a vector (which is something that can be used to hold a direction and an intensity).

Real life is not 2 dimensional, therefore, in my mind anyway, a wind vector should have x, y and z parameters. --AaronEJ (discuss • contribs) 00:59, 29 July 2015 (UTC)

Formal definition
Well, saying that a function can be formally defined as, or in terms of, a concept of a "rule" is simply wrong on several counts.

First, a formal definition can only use the term "rule" if this term has already been formally defined for denoting a mathematical object, and in a way which does not rely on the concept of a "function" already. E.g. you cannot rely on "computation", since that formal concept relies on Turing Machines, and these are built out of functions.

Second, the image is misleading. If two functions f and g from R to R are defined by f(x)=x+x and g(x)=2x, then f and g are defined by different rules; one relies on the addition function, the other on the multiplication function. So if functions are the same as rules, then these are distinct functions. However, this is not the view actually taken in mathematics; they are identical functions, since their domains and ranges are identical, and f(x)=g(x) holds for all x in the common domain. Moreover, you will not be allowed to say that the derivative of the function that maps x to x^2 is equal to either of x+x or 2x, since derivative is defined by yet another rule, which involves a limit of a differential quotient.

It isn't intuitive either. When I stumble out of bed in the morning, find a pair of two identical looking socks, and master to put one on each foot, I know that in the end I will have successfully determined a function from the pair of socks to my pair of feet, and yet there is no rule whatsoever involved for determining this function. For similar reasons it doesn't even work well to explain informally that a function is something like a "rule". It blocks students from making any sense of the Choice Axiom, and this is not fortunate for those who want to go on to study Analysis and measure theory. 119.71.71.131 (discuss) 06:02, 4 October 2015 (UTC)