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March 25

Parent function

What is a parent function? Admittedly, it's been a while since I got my math degree, but I don't remember ever hearing the term. The only page I found on WP that mentions it is Isocline, but this fails to explain what it is. Mathworld (Wolfram.com) fails to have an entry for it, either. — Loadmaster (talk) 03:09, 25 March 2013 (UTC)[reply]

What is the context? —Bkell (talk) 03:11, 25 March 2013 (UTC)[reply]

My daughter's 7th grade homework (ha!), in which she has to determine the parent function of several functions, e.g., a set of parabolas. I just found this video, which explains that it's essentially the simplest of a family of functions, e.g., for n-degree polynomials it is y = xⁿ. WP ought to have an article for this. — Loadmaster (talk) 03:16, 25 March 2013 (UTC)[reply]

I just added a stub article for parent function. — Loadmaster (talk) 03:34, 25 March 2013 (UTC)[reply]

Hmm. I've never heard that terminology used before. I watched the video you linked, and I understand the concept in the examples that were given, but the "definition" given there is not very rigorous, and it's hard to extend it beyond very basic families of functions. I imagine the point of all of this is to lead into the graphing of functions via transformations, such as vertical and horizontal shifting and scaling (and I can't find a Wikipedia article about this, either). If that's the goal, then the idea is just that there are a handful of "fundamental" functions, like y = x, y = x², y = √x, and y = |x|, the graphs of which the student should have memorized; and then graphs of related functions can be found by transforming these "fundamental" graphs. So the term "parent function" seems to mean just "the function in this list of 'fundamental' functions that most closely resembles the given function." With that definition, though, for any list of "fundamental" functions there are many functions that have no parent function at all. For example, what should the parent function of f(x) = x³ + 2^x be?

I don't think we should have a Wikipedia article about this unless we can find a solid, rigorous definition of the term from a reliable source. —Bkell (talk) 03:53, 25 March 2013 (UTC)[reply]

Here's another question to ponder: According to the current definition in the parent function article ["a parent function is the simplest function of a family of functions that preserves the definition (or shape) of the entire family"], why should y = x⁴ be the parent function of y = x⁴ − 10x² + 9? The graphs of those functions have rather different shapes. —Bkell (talk) 03:58, 25 March 2013 (UTC)[reply]

I'd think it should be y = x⁴ - x², if the rule is to drop the coefficients and constants. StuRat (talk) 04:10, 25 March 2013 (UTC)[reply]

Why should that be the rule? That contradicts what the parent function article currently says: "For the family of n-degree polynomial functions for any given n, then, the parent function is simply xⁿ." This all seems very arbitrary to me. —Bkell (talk) 04:16, 25 March 2013 (UTC)[reply]

It also contradicts the video that Loadmaster posted, which is the only source for the article at the moment. That video clearly states, for example, that the parent function of y = −x² + 10x − 20 is y = x². —Bkell (talk) 04:23, 25 March 2013 (UTC)[reply]

Based on the [lack of] 'context' given I suppose the meaning of function x → x² being a 'parent' function for all ax² + bx + c might be that each function of the latter family can be obtained from the 'parent' function by a shift in its argument and value space or scaling. That however strongly limits an application area of the 'parent function' notion, because no such 'parent' function exists for polynomials of 3rd degree or higher. Possibly the notion is used only at 7th grade...? --CiaPan (talk) 06:43, 25 March 2013 (UTC)[reply]

Let's not take anything that the parent function article says as established fact, because I just wrote it. It's based on the video I found. There do not seem to be many authoritative sources out there in net-land to say anything definitive about the subject. (Further discussion about this should continue on the article talk page.) — Loadmaster (talk) 17:29, 25 March 2013 (UTC)[reply]

A Question of Convergence

I am unable to calculate $\lim _{n\to \infty }{\int _{0}^{\infty }{e^{-e^{x^{n}}}}}$ . Even with all the aid of mathematical software, I can only understand its behaviour on the interval [0, 4], but whether afterwards it converges asymptotically towards a certain value, or whether -on the contrary- it continues to grow -slowly but steadily- towards +Infinity, is a mystery to me... Could anyone please help me ? — 79.113.235.6 (talk) 06:12, 25 March 2013 (UTC)[reply]

Dollars to donuts this thing converges to e⁻¹. I don't have a rigorous proof of that claim, but look at the graph of exp(−e^xⁿ) for a few values of n, and you can see what's going on. —Bkell (talk) 06:27, 25 March 2013 (UTC)[reply]

Okay, here's an outline for how to deal with this thing. Split the interval of integration into two subintervals, [0, 1] and [1, ∞). For 0 ≤ x ≤ 1, show that

e^{-1}(1-x^{n})\leq e^{-e^{x^{n}}}\leq e^{-1}

, and for 1 ≤ x < ∞, show that

0\leq e^{-e^{x^{n}}}\leq e^{-e}x^{-n}

. Now use something like the squeeze theorem to show that the integral on [0, 1] approaches e⁻¹, and the integral on [1, ∞) approaches 0. —Bkell (talk) 06:55, 25 March 2013 (UTC)[reply]

Is not the following simpler? :

\lim _{n\to \infty }{\int _{0}^{\infty }{e^{-e^{x^{n}}}}}dx=\lim _{n\to \infty }{\frac {1}{n}}{\int _{0}^{\infty }{\mu ^{1/n-1}e^{-\mu }}}d\mu =\lim _{n\to \infty }{\frac {1}{n}}\Gamma (1/n)=\lim _{n\to \infty }{\frac {1}{n}}{\frac {\pi }{\Gamma (1-1/n)\sin \pi /n}}=1

Ruslik_Zero 19:43, 26 March 2013 (UTC)[reply]

I'm afraid in this case

x(\mu )={\sqrt[{n}]{\ln \mu }}

, whose derivate is

dx(\mu )={d\mu  \over {n\cdot \mu \cdot {(\ln \mu )^{1-{1 \over n}}}}}

, as opposed to what you wrote. Also, the new limits of the interval of integration would have to be 1 and Infinity instead of 0 and Infinity. — 79.113.221.135 (talk) 20:40, 26 March 2013 (UTC)[reply]

Resolved

Loss of information

What is it called when information about a number of function is lost throught some sort of manipulation, which makes the manipulation irreversable. For example, negative one squared equals one, the reverse: taking the square -root of one, does not equal negative. By performing the square function, you lose the negatie sign. Plasmic Physics (talk) 21:41, 25 March 2013 (UTC)[reply]

The function is not injective. —Bkell (talk) 22:06, 25 March 2013 (UTC)[reply]

Does not injective include functions such as arcsin(sin(x)) for |x| > π? Plasmic Physics (talk) 22:13, 26 March 2013 (UTC)[reply]

The sine function is not injective. A composition of functions f(g(x)) cannot be injective if g(x) is not injective. —Bkell (talk) 23:36, 26 March 2013 (UTC)[reply]

Is non-injectivity the only source for fallicious proofs? Plasmic Physics (talk) 05:54, 27 March 2013 (UTC)[reply]

No, of course not. There are many different ways to write a fallacious proof. See Mathematical fallacy. —Bkell (talk) 12:18, 27 March 2013 (UTC)[reply]

Integer-only Mathematics

Is there a specific name for mathematics that only uses integers and all functions take integers and return integers? — Preceding unsigned comment added by 71.204.230.66 (talk) 23:44, 25 March 2013 (UTC)[reply]

See "Number theory".—Wavelength (talk) 00:11, 26 March 2013 (UTC)[reply]

Computability theory also deals with functions whose domain and range are the natural numbers, although its primary focus is the question of which of these functions can be computed by algorithms, rather than the kinds of questions that are usually asked about functions in, say, mathematical analysis. The calculus of finite differences is roughly an analogue of calculus in the domain of the integers. —Bkell (talk) 05:38, 26 March 2013 (UTC)[reply]

Modular arithmetic. StuRat (talk) 05:57, 26 March 2013 (UTC)[reply]

Related fields that deal with mathematical objects that can be indexed by the integers are discrete mathematics and combinatorics. Gandalf61 (talk) 08:45, 26 March 2013 (UTC)[reply]

The adjective Diophantine comes to mind, though I’m not sure it can be used for non-algebraic functions.—Emil J. 13:21, 26 March 2013 (UTC)[reply]

See "Category:Number theory".—Wavelength (talk) 15:28, 26 March 2013 (UTC)[reply]