How to describe functions (math)

[JJChicken]

What would be a standard set of questions to ask for each function that would describe its behaviour reasonably well under most circumstances?

I'm working on a list at the moment, but I might have missed out on a few:

* Behaviour as x --> +/- infinity
* Is the function odd/even? Is it symmetric about any line/function
* Monotone increasing/decreasing?
* How many turning points does it have
* What happens to the slope as x changes - does it increase as x increases (etc.)
* Behaviour of higher derivatives?
* Any asymptotic behaviour?
* Comparisons to common functions (e.g. similar to e^x but with a slope that increases much faster)
* When is it positive / negative
* Unique roots?

I'm trying to create a standardised list to apply to functions of ONE variable.

 

[CycloWizard]

That's a pretty good list, but I'm not sure what the overall point is in the end. Are you just trying to create some sort of searchable function database where you'll pull up functions that have a specified set of properties? You might add additional information about convex domains and the like as that is an often-used property.

 

[JJChicken]

That's a pretty good list, but I'm not sure what the overall point is in the end. Are you just trying to create some sort of searchable function database where you'll pull up functions that have a specified set of properties? You might add additional information about convex domains and the like as that is an often-used property.

Thanks! Convexity is a very useful property to consider. A little background to this list, I'm trying to write a mini guide to explain math concepts at an intuitive, concise yet somewhat rigorous manner for finance professionals. I'm thinking its a good idea to standardise the properties of functions so that the reader can develop a mindset where he or she is actively thinking through a checklist of these properties when coming across new functions for the first time.

I'll probably dumb down the list in the end, I think the points on derivatives and turning points are bit too advanced for a standardised list - if anyone needs this information, they can perform the necessary derivations.

 

[Hacp]

Thanks! Convexity is a very useful property to consider. A little background to this list, I'm trying to write a mini guide to explain math concepts at an intuitive, concise yet somewhat rigorous manner for finance professionals. I'm thinking its a good idea to standardise the properties of functions so that the reader can develop a mindset where he or she is actively thinking through a checklist of these properties when coming across new functions for the first time.

I'll probably dumb down the list in the end, I think the points on derivatives and turning points are bit too advanced for a standardised list - if anyone needs this information, they can perform the necessary derivations.

The best way to explain is by examples. Let them discover themselves what functions are.

 

[DrPizza]

Your list is a little redundant. As mentioned/added, concavity. You already mentioned that in your list though: "what happens to the slope as x changes." If the slope is increasing, the function is concave up. If the slope is decreasing, the function is concave down. Behavior of higher derivatives - you would most likely be referring to the second derivative, which again is concavity.

Missed from the list though:
Inflection points (where concavity changes from one sign to the other)

More important things missing from the list: domain, range, and continuity.

 

[JJChicken]

Your list is a little redundant. As mentioned/added, concavity. You already mentioned that in your list though: "what happens to the slope as x changes." If the slope is increasing, the function is concave up. If the slope is decreasing, the function is concave down. Behavior of higher derivatives - you would most likely be referring to the second derivative, which again is concavity.

Missed from the list though:
Inflection points (where concavity changes from one sign to the other)

More important things missing from the list: domain, range, and continuity.

Thanks Doc

 

[balsa model]

for finance professionals

* periodicity