A question for the math experts - Crazy graphs and their equations

[MattC]

Is it possible to prove that an equation exists to describe any possible graph?

Specifically, can an equation exist which yields a graph that, for instance, is identical to a sine wave with a certain magnitude for the x= a to b, then some random exponential graph from b to c, and then maybe some holes (limits) and such? I'm not asking about any specific graph ... I'm just wondering if, short of graphs with infinite, non-repeating patterns, everything that can be drawn on a graph has an equation that corresponds to it.

 

Thank you in advance for your thoughts!

 

If you're curious as to why I ask this, my girlfriend had a question for me about a homework question for calculus. The question regarded limits and it went a little like this:

If the limit as f(x) approaches 5 is 2, can f(5)= 4? The answer is yes, because the graph may approach the limit, but have a hole there and leap to another part of the graph, whereat it continues on it's path. I said no, thinking that the only way to get a limit in an equation is to have some undefined point, whereat a certain x value yields an undefined function. If the function is undefined at that value, it cannot, by definition, have a value (4, in the case of my example of f(5)=4).

 

My best guess is that the only reason that the answer is "yes" is because graphs can have range-specific functions, e.g.

for x values between -infinity and 0, f(x) = x + 1

for x values at 0 or above 0, f(x) = x + 3

[Dave]

The answer is yes, for far easier reasons than the one described above (although they are correct). Essentially, a graph is one particular way of describing a function. In fact, the definition is explicitly dependent upon a function - the graph of a function [math]f : A \to B[/math] is defined by:

 

[math]G(f) = \{ (x, f(x)) \ | \ x \in A \}[/math]

 

So in other words, without a function the term 'graph' is meaningless.

 

On the other hand, if you're saying "if I draw a line on a piece of paper, can I find a function which represents it", then the answer is 'not necessarily'. If that line is self-intersecting, then there is no possible way that it could represent the graph of the function, since the function would not be well-defined.

[Rasori]

On the other hand, if you're saying "if I draw a line on a piece of paper, can I find a function which represents it", then the answer is 'not necessarily'. If that line is self-intersecting, then there is no possible way that it could represent the graph of the function, since the function would not be well-defined.

 

In other words, if a line drawn passes the "vertical line test," it is a function and therefore an equation can technically be made to represent it? I say technically because such equation may be different for an infinite number of ranges, but it still exists.

[Dave]

In other words, if a line drawn passes the "vertical line test," it is a function and therefore an equation can technically be made to represent it? I say technically because such equation may be different for an infinite number of ranges, but it still exists.

 

Essentially, yes

[river_rat]

The catch here though is that when most people say "can you write down an equation for this graph" they usually mean "can you write down an equation using only finitely many additions, multiplications, exponents and compositions of elementary functions for this graph".

 

For this question the answer is sadly no, in fact almost all continuous graphs are not of that type.

[Tom Mattson]

Is it possible to prove that an equation exists to describe any possible graph?

 

As others have remarked: No, it isn't.

 

Specifically, can an equation exist which yields a graph that, for instance, is identical to a sine wave with a certain magnitude for the x= a to b, then some random exponential graph from b to c, and then maybe some holes (limits) and such?

 

Sure, you can do that. Just define the function in a piecewise manner, as follows.

 

[math]f(x) = \left\{\begin{array}{cc}sin(x) & 0 \leq x \leq \pi \\e^x & \pi < x \leq 2\pi \\ 2 & x>2 \pi,x\neq 3\pi\end{array}\right\}[/math]

 

The above function is a sine wave from 0 to pi, an exponential function from pi to 2pi, and a constant function from 2p onwards, with a hole at 3pi.

 

edited to add:

 

OK, can one of you computer geeks tell me how to write a piecewise-defined function in LaTeX without putting the right curly brace there? If I leave it out then the rendering software says that I have a LaTeX error.

[Cap'n Refsmmat]

[math]

f(x) = \left\{\begin{array}{cc}sin(x) & 0 \leq x \leq \pi \\e^x & \pi < x \leq 2\pi \\ 2 & x>2 \pi,x\neq 3\pi\end{array}\right.

[/math]

Click to see what I did. (Go Google!)

[the tree]

OK, can one of you computer geeks tell me how to write a piecewise-defined function in LaTeX without putting the right curly brace there? If I leave it out then the rendering software says that I have a LaTeX error.

With the cases thingumy.

[math]f(x) = \begin{cases}

sin(x) & 0 \leq x \leq \pi \\

e^x & \pi < x \leq 2\pi \\

2 & x>2 \pi,x\neq 3\pi

\end{cases}[/math]

With TeX it's always important to write what you're saying, not what what your saying should look like.

 

edit Cap'n beat me to it, but my post was still better.

[jcarlson]

Is it possible to prove that an equation exists to describe any possible graph?

Specifically, can an equation exist which yields a graph that, for instance, is identical to a sine wave with a certain magnitude for the x= a to b, then some random exponential graph from b to c, and then maybe some holes (limits) and such? I'm not asking about any specific graph ... I'm just wondering if, short of graphs with infinite, non-repeating patterns, everything that can be drawn on a graph has an equation that corresponds to it.

 

Thank you in advance for your thoughts!

 

If you're curious as to why I ask this, my girlfriend had a question for me about a homework question for calculus. The question regarded limits and it went a little like this:

If the limit as f(x) approaches 5 is 2, can f(5)= 4? The answer is yes, because the graph may approach the limit, but have a hole there and leap to another part of the graph, whereat it continues on it's path. I said no, thinking that the only way to get a limit in an equation is to have some undefined point, whereat a certain x value yields an undefined function. If the function is undefined at that value, it cannot, by definition, have a value (4, in the case of my example of f(5)=4).

 

My best guess is that the only reason that the answer is "yes" is because graphs can have range-specific functions, e.g.

for x values between -infinity and 0, f(x) = x + 1

for x values at 0 or above 0, f(x) = x + 3

 

 

 

In regards to your second question, about your girlfriends homework, While a function CAN have a limit at an undefined point, it most certainly can also have a limit at a defined point, and the limit L at point c need not equal the value of f©. Indeed, the definition of a limit makes no mention of the value of f© at all!

 

The formal definition of a limit [math]\lim_{x\to c} f(x) = L [/math] is for every [math]\epsilon > 0[/math] there exists [math]\delta > 0 [/math] s.t for all [math]x[/math], [math]0 < |x - c| < \delta[/math] implies [math]|f(x) - L| < \epsilon[/math]. The value [math]f©[/math] only starts to matter when discussing continuity at c.

[CPL.Luke]

you can always perform a fourier transform in order to get an exact equation for the graph, (although it may only be in the limit, as you would have to approximate the integrals)

[river_rat]

you can always perform a fourier transform in order to get an exact equation for the graph, (although it may only be in the limit, as you would have to approximate the integrals)

 

Fourier transforms only work for a small percentage of functions CPL.Luke, so i'm not sure how transforming to the frequency domain would help you arrive at any equation in general. What did you have in mind?