Calculus Query

[Karthik]

Originally Posted by phrontist

I meant (and said) inverse, not reciprocal. This isn't an assignment, just something that occured to me during my endless and tedious BC Caluclus (anyone else have Early Transcendentals?) homework.

I've been trying to approach the problem visually, the inverse of a function being that function "mirrored" about the line y=x. I think perhaps this is needlessly painful, I'm going to try some algerbraic manipulation of the definition of the derivative...

My guess is that no such function exists, but I wonder if there is an elegant proof of that.

The following fact will lead you in the right direction.

Let f be a function that is differentialable on an interval I. Suppose that f has a defined inverse function, called g.

Using the defintion of an inverse, and the chain rule, it can be shown that

g'(x) = 1/f'(g(x)), where f'(g(x)) != 0

This should help with you algebraic manipulation.

[sciguy125]

You know what? I was tempted to try this problem, but my hatred of calculus overcame me while I was working some stuff out. Calculus tends to have more letters than numbers. This is math, not grammar... These letters also tend to not be in the English alphabet. Again, this is math, not a foreign language.

Today's episode of Sesame Street was brought to you by... The number e ...And, the letter mu.

There also tends to be odd quirks. Take Gabriel's Horn. How can you possibly have something with an infinite surface area, infinite cross-section, but a finite volume!? And the volume isn't just finite, it happens to be a very specific number/non-English letter: pi.

Yes, I realize the power of calculus. I also respect it. Seeing it work in physics still amazes me. But, I just don't like it.

[dubious elise]

This one is seriously making my head hurt. I recollect hearing a similar question proposed but my brain is too fuddled to remember the question in its entirety.

I say if 3rd semester Calc doesn't cover it, then it just isn't worth the headache.

...not yet, at least

[JamesBrown]

Ok well I cannot find a way to definitly prove it but I can prove that this is never the case in the following cases.

x^n where n is positive
x^n where n is negative
all linear functions,
All absolute value functions
x^n when x is not an integer
All trig functions
All step (integer) functions of x

Thats all I have time for now, Back to AP chem.

[Denman]

lets call f^-1(x) the inverse and (f(x))^-1 the recipricol
anyway
Surely if you have f(x) which has powers in, when you differentiate it the power will decrease by one (unless your using e^x)
reflecting it in y=x inverts the power
(eg f(x)=2x^2+3 f^-1(x) = (x-3)^(1/2) / 2
as such differentiating it only lowers the power by 1
so you can't use integers as when you lower the power it won't be afraction
if you want to try fractions, a/b - 1= b/a which the only thing that that has an equal distance from 1 on either side is 1/2 which would give 3/2 -1 (not equals) 2/3 so that can't work
so not fractions

i'm wondering maybe something hyperbolic or complex?

[EricH]

C'mon, guys, I haven't had calculus yet and I can tell you that there is a function like this. A function's inverse is its reflection in the line y=x, right? A function is any line or relation where any input has one and only one output, right? So a straight diagonal line is a function, correct? Now, the only function that will equal its own inverse has to have a slope of -1. So, any function whose equation reads something like f(x)=-x+b with any value for b will be its own inverse. Was that so hard?

[Manoel]

Originally Posted by EricH

C'mon, guys, I haven't had calculus yet and I can tell you that there is a function like this.

Well, you didn't read the question with enough attention. Phrontist wants a function such that its derivative is equal to its inverse, not the function itself.

Originally Posted by EricH

A function is any line or relation where any input has one and only one output, right?

No, not at all. My previous example of y = x^2 would not be a function if you were correct. You described a bijective function, but there are also injective and, the ones that disproves you, surjective functions.

Originally Posted by EricH

So a straight diagonal line is a function, correct? Now, the only function that will equal its own inverse has to have a slope of -1. So, any function whose equation reads something like f(x)=-x+b with any value for b will be its own inverse. Was that so hard?

Yeah, that is correct if you disregard the derivative part of the problem.

[EricH]

I stand corrected.

[Denman]

Originally Posted by Manoel

Well, you didn't read the question with enough attention. Phrontist wants a function such that its derivative is equal to its inverse, not the function itself.



No, not at all. My previous example of y = x^2 would not be a function if you were correct. You described a bijective function, but there are also injective and, the ones that disproves you, surjective functions.



Yeah, that is correct if you disregard the derivative part of the problem.

but you can only actually call it a function if its 1-1 in theory...

[Manoel]

Originally Posted by Denman

but you can only actually call it a function if its 1-1 in theory...

Care to elaborate? Not sure if I understand what you wrote...

[EricH]

A 1-1 function (also written as one-to-one) is one that only goes down or up, not both, over its entire length. I don't understand why Denman says you can only call a function if it's 1-1 in theory, but he may know more than I do, which is quite likely. However, y=x^2 is a function, and it's not 1-1.

[Manoel]

Originally Posted by EricH

A 1-1 function (also written as one-to-one) is one that only goes down or up, not both, over its entire length. I don't understand why Denman says you can only call a function if it's 1-1 in theory, but he may know more than I do, which is quite likely. However, y=x^2 is a function, and it's not 1-1.

Well, that's not the definition of a one-to-one function; in general it is, indeed, a consequence of the function being one-to-one. However, it does not apply to every function. For example: y = x^3 would be one-to-one by your definition, but, if you consider its complex roots*, then it is not.

* - For the record, I never had much of a formal, theoretical education on complex numbers, just its practical applications. Therefore, I may be wrong on the above statement.

[Denman]

sorry my bad i meant one-many, as long as its only one on the start so you cant get 2 values of f(x) for the same number.... however when you take the inverse (eg y=x^1/2) you need to limit your domain and so would need to limit the domain/range of the original function anyway