Lets say I plot a “path” (whats the math word for the “line” [not a straight line] of a graph?) for a two dimensional graph, in a rectangular coordinate system, just off the top of my head, without thinking about it. Provided that there are never two y-values for one x-value, will there always be a function (albeit possibly extremely complicated ones) that can yield every value on the graph?
If the function is continuous and single valued there is probably an infinite series of some kind that will fit the curve as closely as needed. On the other hand IANAM
The definition of a mathematical function is any mapping from one set onto another, as long, as you say, that there are never two y values for one x value. So, in that sense, your graph defines a function.
I can just pick a mapping out of thin air,
f(1)=3, f(2)=17, f(-7)=pi, f(9.23)=3.
There you go. I’ve defined a function from the set {-7,1,2,9.23} onto the set {3,pi,17}.
I think what you’re asking, is there a way to express that function in what I think is termed “closed form”, as some combination of functions, such as addition, subtraction, sines, cosines, and so on.
In that case, the answer is, I’m sure, no. I’d have to think about how to prove that, but I would think it would have something to do with Godel, or Church’s theorem.
There are only a countably infinite number of expressions that could be formed by combining a finite number of limited functional components, whereas there are an uncountably infinite number of functions over the Real number system.
Yes. Take a look at Chebyshev polynomials and others. They are used to create functions which will conform to a certain data. The US NAval Observatory used to publish a Computer Almanac which had the coefficients of polinomials used to calculate certain astronomical functions. It had a pretty good introduction to Chebyshev polynomials but I am sure you can find texts or online information
Would be physicist checking in - you actually don’t need anything very complicated to do this. A simple Fourier series can give you a mathematical representation (to arbitrary accuracy) of any function with a finite number of finite discontinuities - that is, a function that doesn’t spend all of its time jumping between +ve and -ve infinity. As long as you didn’t draw anything too weird, it’d be easy to find the appropriate series. Google on ‘Fourier series’ for more information.
It’s interesting to note that it doesn’t need to be, as you specified, a function with only one value of y for each x. It actually has to be a single valued function of some variable - other possibilities include y, r and theta.
This site seems to explain how a Fourier series works quite nicely. I’ll try to simply explain what’s going on…
Every graph/line can be drawn as a bunch of sine (sin) or cosine (cos) graphs all added together, with the more sin / cos graphs added together the more accurately the original line can be reproduced.
From that link (which uses cos), the Fourier series is written as:
x(t) = a0 + a1 cos (wot + q1) + a2 cos (2wot + q2) + … + aN cos (Nwot + qN)
Which translates into English as ‘one number [a0], plus the cosine of a constant [w0] and a phase change*[q1] multiplied by a scalar [a1], plus the cosine of twice the constant [w0] and a different phase change [q2] multiplied by a different scalar [a2]’.
OK I know that sounds complicated but bear with me.
On that site, draw a random line on the white area, change the value of ‘Fourier series coefficents’ to 2 and press calculate. The red line is 3 graphs added together - the first is a constant line [a0 from above], the second is the a1 cos (wot + q1) bit and the third is the a2 cos (2wot + q2) bit. If you press ‘table’ you can see the values of a0, a1, a2 (magnitude) and q0, q1 and q2 (phase change).
Now change the value of ‘Fourier series coefficents’ to 100 and press calculate, the red line should fit your original random blue line (almost) exactly. Now it’s 100 little cos graphs all added together.
So in answer to the OP, yes there is a function (or in this case 100 little ones added together) that can describe a line.
Incidentally, this system is used to record music digitally - remember the table of numbers from that site? Well, to record a sound wave as a series of numbers (which can be binary numbers) all you have to do is write down all the numbers in that table and hey presto!, when you send that series of numbers to a digital player, it ‘draws’ the graph of the sound and transmits it. IIRC, CD quality music uses about 44,000 Fourier coefficents (ie 44,000 little graphs added together) to record the quality it gets.
Thanks everyone. Tuco, that is a great link for the topic. But I almost crashed my computer when I put in 10,000 instead of my intended 1,000 (which is probably still pushing it.)
Your last paragraph is the proof for the conjecture in the first paragraph of what I quoted.
Now here’s a conjecture that might take a while to prove: most functions from R to R are not continuous at any point, and these functions don’t have Fourier representations. Any takers?
Define “most”.
The analogous situation is with continuous functions that are differentiable at at least one point–the set of all such functions is a countable union of nowhere dense sets. However, the metric space there is C(R), and that’s not good enough for this. Let’s say the metric space we’re interested in is R -> R. For those unfamiliar with the notation, C(R) is the set of all continuous functions from R to R, R -> R is the set of all functions from R to R.