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How to introduce concept of function?

The concept of " function" is corner stone of mathematics. Those who had basic idea of " function" can be well versed and will be able to understand concept of calculus: LIMITS, CONTINUITY, DIFFERENTIABLITY.

please give suggestions how to dovelop this idea
Development may be
1)Giving examples: Area of circle is function of radius , and establishing there is a relation between two variables and establishing how one quantity will depend on another quantity
2)Giving idea of taking equation:
a)making table of values:
b)ploting equation graphically
3)Function as a relation
4)Establishing a fact that a  line parallel to y axis cuts the graph of a function at one point.
4)Giving examples of expression which is  not a function
5) with conclusion   etc etc
please suggest the steps to be involved systematically

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Replies to This Discussion

Function machines are still the favorite for me. The metaphor easily establishes concepts such as inverse, composition, iteration, domain, range.

I also like to use software that dynamically connects tables of values, formulas and graphs. Spreadsheet software from Open Office or Google, and GeoGebra, are my favorites.
Use a free software package that allows algebra and geometry to be used concurrently. My favorite package is GeoGebra (www.geogebra.org) I find students learn better when algebraic functions are introduced concurrently with their geometric counterparts, rather than introducing functions in the more traditional serial fashion. GeoGebra has one panel that shows the algebra, while the main panel shows the geometry.
We can also introduce a function in an abstract way then go toward something more concrete.

I tried to do something like that in the attached document.
Attachments:
"If you imagine the two initial coordinates to be values for latitude and longitude, for example, then the Zeta Function returns the altitude for every point, forming a kind of mathematical landscape full of hills and valleys." That's an analogy used by Matt Parker to explain the Zeta Function, a function of two variables: " It's in his article: Win a million dollars with maths, No. 1: The Riemann Hypothesis at Guardian.co.uk.

The Sinc function is often used to demonstrate 2D and 3D plotting software. The landscape analogy is notable in its 3D representation, where it appears as a hill surrounded by circular moats and ridges.
y is a function of x when:

for each allowed value of x, there is only one possible answer for y.

Repeat that about 50 times and I find that students have a pretty decent understanding. I say this when looking at tables of values to determine if they are functions, I say this when looking at graphs and using the vertical line test, and I say this when looking at equations.

It also introduces the idea of domain by use of the phrase "allowed values of x". The domain of a function is the set of all those allowed values of x.

Hope this helps.
To clarify my comments, I do not literally repeat the definition 50 times consecutively, as in a mantra.

But at each relevant moment, I repeat the definition, perhaps stating it slightly different, perhaps encouraging the class to 'fill in the blank' when I stop.

I talk about why the vertical line test works for determining if a graph is a function. Drawing a vertical line is picking a value for x. The number of times it crosses the graph is the number of corresponding values of y. If it crosses more than once, that means more than one answer for y. I count that as repeating the definition.

I am also quite repetitious when teaching rational expressions. At least once in each problem I do on this material, I will state or get one of my students to state (by asking a question with this as the answer): "You cannot cancel out terms. You can only cancel out factors" or some variation thereof. I used to have a terrible time with teaching this material. But for the last few years, repeating that rule ad nauseum (for me) really seems to make a difference.

Full Disclosure: I teach mostly 'developmental' math at a community college. I'm not sure under what conditions this 'repetition method' would work.
Thanks to all for guiding.
can we use geogebra in bringing concepts like, limit , differentiability, continuity and Integration?.
Also I need suggestion that How we can link, function, limit, continuity , differentiablity and Integral of a function.
Is there any tutorial for geogebra for constructing java applets

Christian Baune said:
We can also introduce a function in an abstract way then go toward something more concrete.

I tried to do something like that in the attached document.

Christian Baune said:
We can also introduce a function in an abstract way then go toward something more concrete.

I tried to do something like that in the attached document.
I also use the concept of a machine; I made a quickie starter mathcast to show you what I mean: http://www.screencast.com/t/XLyggaJAXaxn
As I said in video, I don't actually introduce functions, but rather review them to kids who never really got it.
I found this concept works well particularly for
(a) demonstrating domain (done on video)
(b) showing only one y can come out of the machine for each x (vertical line test)
(c) showing that the function is egg -> omelet, but finding root is omelet->egg ...
I would love to see/hear how others present.
Best, Linda
The following example of a circle drawn in a Cartesian coordinate system illustrates some of the points being discussed.
The equation of a circle of radius r is not a function:

This can be rearranged to express x as a square root:

and y as a square root:

Neither of these are (single valued) functions, as there is a positive and a negative square root.
We can construct two functions using the absolute and negative of the absolute, using y as the dependent variable out of arbitrary convention:

Defining the radius as r=1, we can plot these two functions over the range x=-1 to x=1, using a free plotting program for Microsoft Windows, such as http://www.padowan.dk/graph/ by Ivan Johansen.

Equations were rendered using http://www.numberempire.com/texequationeditor/equationeditor.php.