In Mathematics, what is a function?

I had two ways to think of a function , for my students :

Certainly its a one y per x idea, and thus its also a Vertical line testable relationship , as described above . Here are two other ideas :

a) The function machine idea :

see http://www.shodor.org/interactivate/activities/Fun...

The function is a set of rules ( on the side of the machine ) and it gives a process of what to do to any number ( or variable ) that you choose to chuck into the machine .

eg if the function is 2x + 5

chuck in a 4 : 2(4) + 5 = 13 <-----the machines output

throw in a (2t): 2(2t) +5 = 4t +5

add a firetruck : output is 2 ( firetruck) + 5

and so on .....

b) The Calculator Idea Model :

All the buttons on most ( simple ) calculators MUST give just one number output because its a simple screen, with only one output possible. So this model is that if its on your calculator it MUST be a function :

Is Sin(x) a function ? Yes.. it has its own key

is ln(x) a function >> yes, it has a button , and gives just one output for any number chucked in.

Is acrsin(x) = sin^(-1)( x) a function : strictly speaking no, but as it has been placed on most calculators,as a button , its been truncated to force it to be a function.... so yes.

and finally : Is a circle ( x^2 + y ^2 = 16 ) a function ???

Well, does your calculator have a " circle " button ? No ? Then its not a function .

( and you can confirm this and many other non functions by just sketching and seeing how they violate the vertical line rule )

And this nicely extends to parametric, and polar plots:

Is the circle in polar form a " Function " ?

r = 4 or r = 3( 1 - cos ( theta )) each give just one radial point for any theta, so its a polar function ... and the Vertical line test becomes the radial line rule ... any radial line from the origin , outwards passes thru the polar function, just once . And all the parts are on your calculator .

A function is a mathematical expression where there exists only one result for each unique argument provided.

That is to say, that there is only one value of f(x) for each value of x. On a graph, you can test whether or not you're looking at a function using the "vertical line test." If you can draw a vertical line that touches the graph in more than one place (such as you can with a circle), then the graph is NOT a function.

A function is something that you perform on numbers or letters. It's a bit like a verb in a sentence e.g. jump up the stairs 2 steps at a time up a flight of 12 stairs - the act of jumping 2 steps at a time is the function or action you're performing on the flight of stairs.

A mathematical function can be basic arithmetic (e.g. multiply a number and keep repeating until infinity) or it can be something more complex (e.g. convert a Celcius temperature value into a Farenheit value).

A function, in mathematics, is usually written as an equation. It is used to show relationships between a set of numbers and another set of numbers.

Cheers! :D

A function is a set of ordered pairs such that no two ordered pairs have the same first coordinate.

a mathematical correspondence that assigns exactly one element of one set to each element of the same or another set.

Well put simply I suppose its a command.

If say your fuction is f(2x) then the command is 2 multiplied by x. (where x can be any number etc etc )