Talk:Inverse function/Archive 2 Source: en.wikipedia.org/wiki/Talk:Inverse_function/Archive_2

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Archive 1

Archive 2

Usually and most of the time :

${\displaystyle \sin ^{-1}x=(\sin x)^{-1}.\,\!}$

and inverse function of

${\displaystyle f^{-1}(\sin x)=\arcsin x=\mathrm {asin} \,x.\,\!}$

If arcsin(x) is used then sin⁻¹(x)=(sin x)⁻¹.

In calculus ƒ⁽ⁿ⁾, where n is a Roman numeral; with or withouth parentheses; that n with equal validity also denotes the nth derivative of a function ƒ. For instance:

${\displaystyle f^{II}(x)={\frac {d^{2}}{dx^{2}}}f(x).}$

Finally, in case of derivative nth function could be represented, and often and usually is, with an apostrophe. For instance:

${\displaystyle f^{''}(x)={\frac {d^{2}}{dx^{2}}}f(x).}$

All these notations are valid and used.

However, I tried to keep my editing as minimal as possible so I edited only categoric statemant regarding sin(x). Hrvatistan (talk) 02:55, 9 July 2010 (UTC)

I disagree, I have only ever seen sin^{-1}x used to mean arcsin x, even though it is inconsistent with sin^2 x. Where have you seen it used to mean (sin x)^{-1}? I also haven't seen e.g. f^3 without brackets for derivative, presumably because it already has two meanings (multiplicative power and repeated function application). Quietbritishjim (talk) 00:40, 10 July 2010 (UTC)

I agree with Quietbritishjim. -- Dr Greg  talk  08:41, 10 July 2010 (UTC)

Well, that is the reason why I only edited only categoric statemant regarding sin(x) because it is possible to use it both ways. And since Karl Weierstrass mathematic is all about consistency. If You use arcsin(x) then it is general rule that You don't use sin^{-1} x for the same thing. And I said Roman not Arabic numerals. If You haven't seen something it doesn't mean that something does not exist. Or like Dolph Lundgren said it: "There's an old saying: just because you're paranoid doesn't mean they're not out to get you." Hrvatistan (talk) 16:24, 31 July 2010 (UTC)

Bourbaki uses a different notation for this. His notation suppress ambiguity, but is only seldom used. It places the ${\displaystyle -1}$ exactly on top of the funtion. For a function ${\displaystyle f}$, it should render something like ${\displaystyle {\stackrel {-1}{f}}(x)}$, which avoid confusion with the function ${\displaystyle f^{-1}}$, whose value ${\displaystyle f^{-1}(x)}$ at ${\displaystyle x}$ is ${\displaystyle f(x)^{-1}}$. — Preceding unsigned comment added by 92.104.193.24 (talk) 15:38, 7 November 2011 (UTC)

I know y'all are going to absolutely hate me for this one ... I perfectly well understand the article and think its actually wonderfully written for me (personally). -- but this is really just because I've messed around with math for long enough that I can actually understand it (with occasional references to old text books ...) but its going to be really hard for most people to read through this without having a lot of trouble. Ideas? Possibly an "Introduction to inverse functions" article? Katanada (talk) 05:03, 28 April 2011 (UTC)

"Introduction to" articles are best created very sparingly. I don't think the problem with the lede was actually a technical language issue, just an instance of confusing language. I attempted to clean that up. If your problem was with the rest of the article, please fill out the section parameter to the template. ᛭ LokiClock (talk) 20:49, 16 June 2011 (UTC)

--JayEB (talk) 05:13, 2 November 2011 (UTC)
Below the table of inverses, there's a method for finding an inverse which just solves
the equation to isolate the other variable. But the step of trading places of the two variables is not there.
So the result seems to be just an isolation of the other variable.(the plot of the transformed function is identical to the original)
I think any change should be after some talk.(as long as this huge talk page does not continue to grow exponentially)(Cliff's Notes ?)

The process of finding the inverse is, in fact, isolating the other variable. x & y are usually swapped so that x will always represent an input and y an output, but the x input to the function isn't the same as that of the inverse, whether you switch them or not. f(a)=2a is the same function as f(b)=2b. ᛭ LokiClock (talk) 01:48, 8 November 2011 (UTC)

--JayEB (talk) 19:22, 3 November 2011 (UTC)
[The following is derived from Bronshtein, Handbook of Math, 5th, 2007, at 2.1.3.7. ,
but I substitute my personal notation f ^1-(x) as a cure for the tainted shorthand: f ^-1(x) ]
[ f ^[-1](x) is too cumbersome, and f ^-(x) is too close to f '(x) ]
The functions y = f(x) and y = f ^1-(x) are inverse functions of each other, for the domains that are functions.
Starting with y = f(x), to get the explicit inverse, you exchange x and y in the expression,
which gives you x = f(y), and solving for y gives the explicit form y = f ^1-(x) .

The plot of y = f ^1-(x) is a reflection of y = f(x) across the line y = x .
Examples of inverse functions: y = x ² has the inverse y = f ^1-(x) = x ^1/2
y = f(x) = e ^x has the inverse y = f ^1-(x) = ln x
y = f(x) = sin x has the inverse y = f ^1-(x) = arcsin x

Nope, it's irrelevant. Exchanging the variables is purely syntactic and totally unnecessary, because a function f(x)=x^2 is the exact same function as f(y)=y^2. Variable names don't matter here.--greenrd (talk) 23:41, 6 November 2011 (UTC)

--JayEB (talk) 06:48, 7 November 2011 (UTC)
y = f(x) = x^2 is a vertical arch symmetric about the y axis.
x = f(y) = y^2 is a horizontal arch symmetric about the x axis.
The axes are x-horizontal and y-vertical for both.

No, you are talking about a 2-dimensional plot of a function, which is not the essence of a function, it's just a drawing of it. All you've demonstrated is that the same function can be plotted in more than one way. Conventionally, the dependent variable is usually on the vertical axis and the independent variable on the horizontal. If you plot a function with a log-scale vertical axis, that doesn't make it a different function.--greenrd (talk) 19:38, 7 November 2011 (UTC)

You can also turn the piece of paper sideways, and then the x axis runs up & down. You haven't inverted the function by doing that unless you cross out x & y and switch them. ᛭ LokiClock (talk) 01:48, 8 November 2011 (UTC)

--JayEB (talk) 06:25, 8 November 2011 (UTC)I will assume you are not putting us on. A function in a real-number cartesian X-Y field is properly
represented by a plot of the function on X-Y coordinates. And its inverse, whether it's also a function or not
is properly represented by a plot of that inverse. One problem in math is that folks don't plot things early on.
Your statement carries a burden to prove that the "essence" of such a function is not properly represented in a plot of the function.
We are not dealing with imaginary numbers here. That's here: http://en.wikipedia.orghttps://demo.azizisearch.com/lite/wikipedia/page/Imaginary_number#Geometric_interpretation

The real numbers aren't an X-Y field. X and Y are labels for the 1st & 2nd dimensions, simple conventions. Cartesian coordinates can use any labels, or none. The Z and Y axis switch roles frequently - either can be used for depth or height. Properly, x, y, and z are just variables. What they usually represent is "a number in the real numbers," so that when the solution to an equation is x, the solution means "any number in the real numbers." When you say f(x)=x, you can interpret f as something with an infinite number of values. It has a solution in terms of x, which will give f as many solutions as their are real numbers. Since the x is the same on both sides of the equation, the solution corresponds to the value in R. When a function depends on two variables, both defined as "any number in R," the symbol being the same on both sides of the equation is what distinguishes the two "any numbers." But you don't label the numbers by what order you pick them in for the equation, nor does picking labeling them give them any sign of input or output - the output of the function is strictly evaluation of f(x). It's the convention on paper that makes y=f(x) for whatever f you happen to be plotting. ᛭ LokiClock (talk) 12:15, 9 November 2011 (UTC)

"All you've demonstrated is that the same function can be plotted in more than one way."
These are not the "same function"; they are inverses of each other. And X = Y² is not even a function;
its a relation, since there are two Y's for each positive X.
Consider the Exponential function Y =e^X, and its inverse X=e^Y (the Log function).

That's a mistake. It's easy to misinterpret the vertical line test in that way. What it actually says is that the true inverse of x², ±sqrt(x) which is the same as ±sqrt(y) and ±sqrt(a), isn't a function, because it maps multiple outputs (±sqrt) to one input (x, y, or a). x=y² isn't the inverse of y=x², it's the same function on different variables. When you plot x=y² without relabeling the axes, you're switching the input/output ROLE of x and y, and the function is unchanged. Then you actually perform the "vertical line test" using a horizontal line - so take the name with a grain of salt. ᛭ LokiClock (talk) 12:21, 9 November 2011 (UTC)

"If you plot a function with a log-scale vertical axis, that doesn't make it a different function."
We are in full agreement with the World on that.

"You can also turn the piece of paper sideways, and then the X axis runs up & down.
You haven't inverted the function by doing that unless you cross out X & Y and switch them."
We are in full agreement with the World on the first part; but re-labeling the axes would not change
the nature or operation of any planar real-number function, nor its inverse.

It does, because by changing the graph in that manner you're actually changing the function you're graphing. On your graph, you've chosen x for input and y for output. So when you relabel the axes on the graph of y=x², it becomes a graph of y=±sqrt(x). Geometrically, reflection about the line y=x is equivalent to taking the inverse through syntactic isolation. Try it with a logarithmic & exponential function of the same base. ᛭ LokiClock (talk) 11:53, 9 November 2011 (UTC)

--JayEB (talk) 00:45, 10 November 2011 (UTC)
The proof I asked for ? After reviewing things, I stand by what I wrote.
And re: x = y² isn't the inverse of y = x²
The inverse of f(x)= x² is f ^-1(x)= x^1/2.
The check is: f [f ^-1(x)] = x (the identity line Y = X )
I've had enough of this thread. The previous article section needs work.

I think you need to ask more questions before you decide. You deny that f(x) and f(y) are the same thing, but you haven't explained why they should be different functions. The "identity line" picture also requires consistency of input/output conventions - x must be input to f, and y must be its output, so to reverse it the inverse must take the output as an input - so it must be f^-1(y), not f^-1(x), when testing f^-1(f(x))=x. That means to change the order to f(f^-1(x))=x, f could only be f(y) and f^-1 could only be f^-1(x). By the logic that f(x) is different from f(y), satisfying the opposite version of the identity statement changes the function and its inverse. In other words, if f(x)≠f(y), f(f^-1(x))≠f^-1(f(x)), and therefore x≠x. ᛭ LokiClock (talk) 10:35, 10 November 2011 (UTC)

--JayEB (talk) 02:58, 19 November 2011 (UTC)
I have no issue with the small table of trig inverses, but the main table seems inconsistent, at the least.
For starters, below the table, for f(x)= y = (2x + 8)³, I think f ^-1(x) should be (x^1/3- 8)/2 and f^-1(y) should be (2y + 8)³ .
If not, then specifically why not ?

There's no f^-1(x) to switch to f^-1(y). It says f(x)=(2x + 8)³. So what change are you suggesting exactly? Right now it's consistent in that the variable isolated in the first part is replaced with the inverse function symbol (written as a function of the dependent variable), and if you switched the variables to make both functions functions of x, you're likely to throw people off. ᛭ LokiClock (talk) 23:16, 19 November 2011 (UTC)

--JayEB (talk) 00:18, 20 November 2011 (UTC)
Confused by f^-1(x) ? Then leave it out. I say the f^-1(y) in the article is wrong, and it should be f^-1(y) = (2y + 8)³ .
Specific reasons why not ? Supporting work and references please.

You seem very confused by standard mathematical notation. f(x)= (2x + 8)³ means exactly the same as f(y)= (2y + 8)³, which means exactly the same as f(q)= (2q + 8)³, which means exactly the same as f(β)= (2β + 8)³. You can use any letter you like, it doesn't change the meaning. By the way, please put your signature at the end of your message, not the beginning. -- Dr Greg  talk  02:00, 20 November 2011 (UTC)

Specific reasons why not ?. f ^-1(y)=(y^1/3-8)/2 is reversing the input variable, not a function inverse.
Plots of the inverses mirror across X=Y and compose to X. You seem to be confused.--JayEB (talk) 02:44, 20 November 2011 (UTC)

f⁻¹(y)=(y^1/3−8)/2 means exactly the same as f⁻¹(x)=(x^1/3−8)/2, which means exactly the same as f⁻¹(t)=(t^1/3−8)/2, which means exactly the same as f⁻¹(φ)=(φ^1/3−8)/2. You can use any letter you like, it doesn't change the meaning. All of those are correct answers for f⁻¹ in this case. If you write y=f(x) then you would write x=f⁻¹(y), but if you are simply talking about f(x) with no mention of y, then you are free to use any letter you like for the inverse. -- Dr Greg  talk  14:33, 20 November 2011 (UTC)

Letters are not the point. I have to think that you just don't understand the issue.
Looking for supported comments by those who do. --JayEB (talk) 16:58, 20 November 2011 (UTC)

It's true I don't understand the issue of why you find this so confusing. It's very simple. If f(p)= (2p + 8)³ then f⁻¹(q)=(q^1/3−8)/2 and you can replace either or both of p and q by x, y, or any letters you like. This is very standard mathematical terminology. -- Dr Greg  talk  17:27, 20 November 2011 (UTC)

I don't understand your point, either. Can you write out the original text, and what they'd become after you changed them? f^-1(x) isn't confusing, it doesn't exist. So I don't know why you're saying it should become something. The answer to your question may in fact be that there's no reason why not, or that it's merely a matter of pragmatism. But "why not" isn't an argument, "why" is. But first... what? ᛭ LokiClock (talk) 17:35, 20 November 2011 (UTC)

I need a math notation editor. I uploaded a pdf to avoid the grief of this editor:

--JayEB (talk) 21:30, 20 November 2011 (UTC)

You took f({0, -2}) to get {512,64}, so you have to take f^-1({512,64}), not f^-1({0,64}) to get {0, -2}. That's why it's written f^-1(y), because you're supposed to set y=f(x) to get x back. ᛭ LokiClock (talk) 21:48, 20 November 2011 (UTC)

Are you sure you aren't confusing inverse relation and inverse relationship? Relations are like functions that don't have to pass the vertical line test, so ignore those terms for the purposes of the conversation. You're saying y=e^x is the inverse of x=e^y, but again, it merely plots the inverse of x=e^y, meaning you get the same plot as taking x=ln(x). Saying X and Y instead of x and y is problematic, because then it looks like you're talking about the sets, which are both the real numbers. e^x sends any real number x to another real number, so it sends X to X. Your problems would be solved if you were talking about the pairs of coordinates, the actual objects on the plot, instead of the input and output those objects are employed to graph. Then, truly, if x and y are coordinates, y=e^x produces the coordinates {(x=0,y=1),(x=1,y=e)}, and x=ln(y) produces the same coordinates - {(x=0,y=1),(x=1,y=e)}, except in one case the 2nd coordinate is derived from taking a function of the 1st, and in the other case the 1st coordinate is derived from taking a function of the 2nd. However, if you use x=e^y you {(x=1,y=0),(x=e,y=1)}, the same coordinates you get from the inverse function on the opposite variables, y=ln(x). It's only when the x and y refer to coordinates that you get this illusion that when functions have the same plot then they're the same, and when they have a different plot they're different. It depends on whether x and y have an input-output role, and whether those roles correspond to geometric realities. When we say f(q) instead of f(x), that's like labelling the horizontal axis q instead of x. Instead, you're using the vertical axis for input and the horizontal axis for output, which is a different situation. Taking the inverse function of the variable with the opposite role will produce the same coordinates. That proves that the resulting coordinates have not changed, not that you didn't have to change both the function and input to do so. ᛭ LokiClock (talk) 22:20, 20 November 2011 (UTC)

Well, technically e^x sends X to X⁺, but it still doesn't help you. The line is a separate set from the input and output sets. Y=X is a true fact - Y=R, and X=R. ᛭ LokiClock (talk) 22:54, 20 November 2011 (UTC)

Put a different way, the 1 on the x axis is the same as the 1 on the y axis, but (1,0) is not the same as (0,1). You can certainly take the coordinates (x,f(x)), and then the coordinates (f(y),y) and get different points, but you can't take the function f(x) and the function f(y) and get different values. ᛭ LokiClock (talk) 22:33, 20 November 2011 (UTC)

I did not wade through the last few notes. I give up. In parting, I mixed up p's and q's in that modified example;
I do not agree; the inverse function should be f^-1(q)= (2q + 8)³ as with the actual case f(x)=(2x+8)³.
A plot of f(x) between(-8,-512)and (0, 512), along with line X=Y makes it clear that the inverse runs through (0,-4) with a very small range in Y. --JayEB (talk) 00:34, 21 November 2011 (UTC)

Okay. I "waded through" your PDF and tried to give a thorough explanation. The reflection you keep using only works when you don't change input and ouput axes before you plot the second function. ᛭ LokiClock (talk) 02:30, 21 November 2011 (UTC)

Dr Greg: You need to prove that the temperature converse was an inverse. You cannot, so leave my correction alone.--JayEB (talk) 22:53, 29 November 2011 (UTC)

Of course I can prove it, and I've added an extra line to do so (though it is really stating the obvious). In your version you gave two different and contradictory formulas for f, which means you are still very confused about the meaning of the symbols f and f⁻¹ (see discussion above).

As for the second edit you made, please go and read the definition of function, and, in particular, the section Specifying a function. There is no necessity to have a "rule" or "correlation relevance".

By the way, I have a Ph.D. in mathematics, what is your qualification? -- Dr Greg  talk  00:01, 30 November 2011 (UTC)

.......................................
I know several math PhD's who don't understand what an inverse function really means.
You transpose to isolate the other variable, then plug it back into the equation; that will ALWAYS return the original variable!
If you plot your "inverse", you will have the same plot on the same axes; that is NEVER an inverse!
But a transposed converse will plot as the same line. My real inverse plots symmetric to F=C.
You don't provide proof, so what allows you to counter proof ? --JayEB (talk) 00:21, 30 November 2011 (UTC)

Interesting that if several PhDs disagree with you, you assume they are all wrong and you are right. The trouble is, you seem to be making up your own definition of "inverse" instead of using the one stated in the article. If we define f and g by f(C) = 9⁄5C + 32 and g(F) = 5⁄9(F + 32), do you not agree that f and g are both functions and that f(C) = F if and only if g(F) = C? That's precisely the definition given in the article (Inverse function#Definitions). Equivalently g(f(C)) = C for every C, which satisfies the alternative definition given at Inverse function#Inverses and composition.

In your version you had f(C) = C 9⁄5 + 32 and f(F) = 5⁄9 (F – 32), which makes no sense at all as you've used the same letter f with two contradictory meanings. I still think you haven't grasped the concept that the definition f(F) = 5⁄9 (F – 32) means exactly the same as f(C) = 5⁄9 (C – 32) or f(μ) = 5⁄9 (μ – 32). -- Dr Greg  talk  01:41, 30 November 2011 (UTC)

I see one problem is that you use this article to defend this article. Try using serious math sources;
several very serious sources; not your memory; not the palaver here.--JayEB (talk) 03:09, 30 November 2011 (UTC)

I think the "I have a PhD" comment has put this discussion a little off track. Let's concentrate on the facts. The title of this article is very clear: it is talking about "Inverse function[s]". The word function is meant in the mathematical sense: objects which accept a number, and using some rule output another number (this is a slight simplification in a couple of ways not relevant here). They are like little black boxes which have a hole you put numbers in and another hole out of which other numbers fall. To reflect this fact, mathematicians sometimes use notation like f: x → 1 + x² (f takes any number x to the new number 1 + x²), which means precisely the same as f(x) = 1 + x² (and the same as f: y → 1 + y², and the same as f(y) = 1 + y²). Functions know nothing about temperatures, or anything else in the real world: all they can see is the numbers we put into them. They don't even know about graphs (we can use graphs to help understand functions, but that's for our own benefit, and it doesn't affect the meaning of "function"), so this stuff about "plotting on the same axes" isn't relevant.

The "composition" of two functions just means that we apply one function then the other to any given number (like connecting up the "out" hole of one function directly to the "in" hole of the next). An "inverse" of a function f is another function g such that when you compose g with f either way round you always get the number out that you originally put in. Note that an inverse function is still a function: it is still a rule for going from one number to another. As Dr Greg correctly says, f: x → 9⁄5x + 32 has inverse function g: x → 5⁄9(x – 32). This is a mathematical fact. If you do not believe this, then I'm afraid that you have misunderstood how functions work. In that case this article (and presumably the one on functions) has failed you; I'm sorry about that, but the talk page isn't the place to get help. The talk page is for people who already understand the subject to discuss improvements. Quietbritishjim (talk) 01:32, 1 December 2011 (UTC)

Too many have learned that a transposed converse is an inverse. They should not be repeating that mistake here.
References can be found to support that, but check Bronshtein, Wolfram, and the consensus references; and ask engineers who actually work with cancelling inverse functions. Words are just words without proof and citations. Prove it to yourself.--JayEB (talk) 02:49, 1 December 2011 (UTC)

Another problem is that you have now introduced an "x" for a binary function with variables C and F.
The "x" does not belong in this example, and it's deceiving when a converse is composed with a function,
since the result looks very much like a definition of an inverse function.--JayEB (talk) 03:13, 1 December 2011 (UTC)

JayEB, before you can understand what "inverse function" means, you must understand what "function" means. The fact you have said '"x" does not belong in this example' shows that you don't know what a function is. A function is just a rule to go from one number to another; the variable name is not part of the function. f is also equal to "multiply by 9⁄5 then add 32". Are you happy to accept this?

You want references? (I note that you aren't giving any!) Any calculus or early analysis book will do, but here's a good one:

• Calculus By Michael Spivak, p 39, "a function is a rule which assigns, to each of certain real numbers, some other real number." Quietbritishjim (talk) 11:47, 1 December 2011 (UTC)

Some function references: https://picasaweb.google.com/112167498780318401154/ScrapbookPhotos --JayEB (talk) 20:15, 1 December 2011 (UTC)

The ones on your first page are good references, and they agree with what I'm saying, especially the bits you've underlined. As they say, the function is a rule that takes a number to a new number. This is why "multiply by 9⁄5 then add 32" is a function, the one we've been talking about all along. And as the first reference says, "functions can be represented by the points (x,y)". Note that it says that graphs are just ways to represent graphs for our understanding; a graph is not a definition of a function.

The reference on you second page is not authoritative. It is from PlanetMath, which is community-written, like Wikipedia. It still has it pretty much right, but is maybe a bit sloppy in places, and almost sounds like the author had a similar misunderstanding to you. (Actually I think they didn't but were just trying to explain the idea behind the function.)

Stick to the references in your first picture, and you should be fine. Quietbritishjim (talk) 21:31, 1 December 2011 (UTC)

Some folks keep writing “advice” that I’ve known since age 12, and use their ‘news-to-me’ in droning, counter-productive palaver. . WP is not a game show. . . The second page you refer to is from Springer’s European Math Society site; I would not cite from Planet Math for these purposes. . .Why was it thought to be from PM ? . . . The idea of function can be as in set math, where tags on individuals can be a plain correspondence. . . . That is unfortunately called a ‘function’, but that’s just a unique inventory list ( maybe that should be called a “set function”). . . Algebraic functions have a rule involved. That rule can have uncertainty within a correlation, as long as that is explicitly included. . .But the subject here is a simple linear equation of f(C) and f(F), with “x” improperly imposed. . .And why should I let some unknown, and somewhat hostile, person tell me which references to believe ?
Too much wasted time here; need fewer words, more proof and citations.--JayEB (talk) 22:51, 1 December 2011 (UTC)
. . . . Another warning to anyone trying to learn from this article . . . .
In the temperature “example”, f(C) is composed with “f^-1(F)” as a “proof”. But for a real inverse, it would be f(C) composed with f^-1(C)
Furthermore: https://picasaweb.google.com/112167498780318401154/ScrapbookPhotos#5681793568547730738 . . . --JayEB (talk) 16:40, 3 December 2011 (UTC)

The word "invertible" in the Wikipedia now redirects to "inverse element", which is something in abstract algebra (e.g. group theory). However, "invertible" should redirect to "invertible function", which is the more familiar concept in analysis and mappings between sets. Then "invertible function" gets redirected to "inverse function", which is sometimes not what we are looking for either. Sometimes we just want to know if a function or a mapping is invertible, and we don't need to know what that inverse IS.
98.67.106.59 (talk) 07:17, 4 August 2012 (UTC)

"This article may be too technical for most readers to understand...."
Yes, yes, it is supposed to be. "Too technical for most readers to understand" is necessary and sufficient. Most readers have not taken or completed "upper-level" high school math, and hence they do not know anything about logarithms, trigonometric functions, inverse trigonometric functions, etc. Often the same applies for exponential functions and functions defined by roots (square roots, cube roots, etc.) or absolute values. They certainly do not know anything about calculus. (Note: a fucntion that has a horizontal tangent line fails to have an inverse there.)
This continues for college students on the freshman level. I have tried to teach students on the level of College Algebra and in Precalculus who thought that all functions were straight lines. Hence f(x) = K/x does not make sense to them because they do not know that its graph is a curve, and furthermore, it has two disjoint pieces to it.
One of my colleagues jokingly said that those students are "instinctively prepared" for calculus, in which all differentialble functions, if you look at them through a powerful-enough microscope, look like straight lines!
98.67.106.59 (talk) 07:38, 4 August 2012 (UTC)

This article needs to be split in two in some way.
Method 1: Split this article into two articles in the Wikipedia.
Method 2: Split this article into two large nonoverlapping sections. Call them Part A and Parr B for the moment.
In either case, Part A will be all about functions of just one real number and their inverse functions. The functions themselves will only have real numbers in their ranges.
In either case, Part B will cover functions and mappings of a lot more generality. The objects in the domains and ranges of the functions can be abstract "points" in the two sets, or to be more specific, pairs of real numbers in both sets, triples of real numbers, complex numbers, quaternions, or functions themselves - whatever you like.
Hence Part A will be strictly on the level of high school students and lower-level college freshmen. Then Part B will be for readers with more background and interest in math, incluing those who have studied multivarible calculus, complex analysis, and whatever.
This ought to cut down on a lot of confusion and arguments - especially concerning people who know something about Part A, but them they want to get argumentative about things in Part B, which is a lot more generalized.
See also the article on the Inverse Function Theorem for some guidance.
98.67.106.59 (talk) 08:25, 4 August 2012 (UTC)

I think this is really very bad terminology. So "log" undoes "exp"? Then how comes that the function exp still exists, since log undid it? One function cannot undo another function, at best, it can undo (sic) the "action" of the other function.

PS: "putting y into the (...) function g " is roughly as bad. Since when are we putting things into a function? C'mon, there's a difference between being understandable and using baby language, esp. in an article about mathematics...

PS2: "Direct variation function are based on multiplication; y = kx. The opposite operation of multiplication is division and an inverse variation function is y = k/x." --- nonsense! (and bad grammar...) The second function does not divide x by k, as opposed to the first which multiplies x by k. (If you are "putting" kx "into" the 2nd function, you get y=1/x!) Maybe the author should first stick to addition, with y=x+a (aka "translation"), then the inverse operation obviously is y=x-a, and not a-x). — MFH:Talk 04:58, 9 September 2012 (UTC)

The lead paragraph: I agree that the first sentence is not as clear as it could be, although I think you're making a bigger deal out of the difference between a function and its effect than is really warranted. Changing "undoes another function" to "undoes the effect of another function" or "undoes the action of another function" would probably better. Perhaps it would be better to put the rigorous definition first and then put this sentence, making it clear that it's an intuitive point of view. But in that case the rigorous definition should be expressed in words instead of symbols, in my view. Either way, it should be corrected - at the moment it only defines a one-sided inverse! Quietbritishjim (talk) 16:25, 9 September 2012 (UTC)

Definition. * is an operation on A, i.e. a*x =b where a, x, b are at A, then x = a^-1*b, if * is conmutative, we can write x = b*a^-1; else x = bºa; at this case º is named inverse operation of *.--201.240.86.150 (talk) 17:42, 26 April 2013 (UTC)

I would like to have here explicit examples ( and discussion) of functions that do not have inverse function. Regards.--Adam majewski (talk) 09:54, 4 January 2014 (UTC)

There is a detailed discussion in inverse function #Partial inverses. The last paragraph of inverse function #Definitions also consider this. Note that section levels currently appear to be broken, with “examples” sections following “Definitions” as ===s (subsections) for no reason; I refer to the last paragraph of “Definitions” proper. Incnis Mrsi (talk) 11:35, 4 January 2014 (UTC)

"Inverses in calculus" should refer to the Fundamental Theorem of the Calculus because the fact that integration and differentiation are the inverse operations of each other is what the relationship between differential calculus and integral calculus is all about.
If Isaac Newton and Gottfried Wilhelm von Leibnitz had not discovered the Fundamental Theorem of the Calculus (of something simpler than it), then we would be in serious trouble in science, technology, and mathematics. Not even mentioning the Fundamental Theorem of the Calculus is a serious lacking.
This article also has other serious problems. I suggest that it needs to be scrapped and redone from scratch.
98.67.96.230 (talk) 23:37, 19 September 2012 (UTC)

Integration and differentiation are inverse operations; this is, however, the article about inverse functions. I fail to see what the history of the calculus has to do with a discussion about inverse functions. A function is a one-to-one mapping from a set A (the domain) to a set B (the range or image). If the mapping is one-to-one, then the inverse exists. If it is one-to-one and onto, the inverse relation is a function.

If you see a "serious problem" then spell it out specifically.

Some calculus may be appropriate in a discussion of inverse functions, but it is not at all necessary to use calculus to define the inverse of a function.

John (talk) 03:44, 24 January 2015 (UTC)

some of it could be simpler, is this the simple English Wikipedia. For example I have just learnt that a way to think of the inverse of a number is 1 divided by that number. Rather than that number divided by 1. So this clearly says what is inverted. The mathematical explanations do not say that in written English. This inappropriate writing style (as found here) give the problems in understanding mathematics. — Preceding unsigned comment added by 141.244.80.133 (talk) 14:20, 15 January 2013 (UTC)

This is not the article about the multiplicative inverse in the field of real numbers ${\displaystyle \mathbb {R} }$, it is about inverse functions. The operation of multiplication in a number field is not particularly relevant here, and an off-topic digression is not a simplification.

John (talk) 00:05, 26 January 2015 (UTC)

Despite their familiarity, percentage changes do not have a straightforward inverse. That is, an X per cent fall is not the inverse of an X per cent rise.

This could do with some rewriting. There is a straightforward inverse to the function "add a fraction p to x", i.e., f(x) = (1 + p) x, and that inverse is "subtract a fraction p / (p + 1)", i.e., "multiply by 1 / (1 + p)". The current text is confusing to someone who knows this inverse, but not the general concept. QVVERTYVS (hm?) 17:55, 5 October 2014 (UTC)

I see your point. You were thinking something like this?

Despite the familiarity of percentages, some find the inverse of a percentage change to be confusing because an X per cent fall is not the inverse of an X per cent rise. To solve this, percentages may be treated as fractions, p = n / 100, where n is the percentage. The inverse of adding a fraction p to x, i.e., f(x) = (1 + p) x, is subtracting a fraction p / (p + 1), i.e., f(x) = [1 - {p / (1 + p)}] x can be simplified to f(x) = [1 / (1 + p)] x.

Should the heading be changed from "non-example" in this case? —PC-XT+ 03:07, 16 October 2014 (UTC)

I suggest that this "non-example" be removed from the article. I find the phrase "non-example" confusing. It could be anything: "Non-example: James Baldwin." There seems to be a general confusion on this talk page about the word "inverse" in mathematics. The scope of the article should be limited. This is an article about inverse functions. "Percentages" aren't functions, they are ratios and numbers. A brief section like this or something similar could be in the article on percentage.

John (talk) 17:05, 24 January 2015 (UTC)

I agree. I don't think it really adds to the subject, which may actually be the reason for the name "non-example." There are plenty of other functions that have what some would think are strange inverses, but they all follow from laws. This one was probably only added due to familiarity. —PC-XT+ 00:27, 25 January 2015 (UTC)

I removed the section as I too thought the same thing as soon as I saw it Belovedeagle (talk) 00:13, 7 April 2015 (UTC)

An example which requires taking cube roots is more complicated than the principle it's supposed to illustrate.

A linear equation would be a more appropriate first example. — Preceding unsigned comment added by 66.35.36.132 (talk) 01:09, 18 March 2016 (UTC)