What Does the Calculus Sentence Not Every Curve... Mean?

[yaya10]

Hello Everyone,

I was reading a calculus book and I have read this sentence that I could not understand.

Can someone please explain it to me.

Your help is highly appreciated.


I am referring to the one starts from (Not every curve ... the given circle).

Thanks a lot.
Yaya

 

[algebrat]

It is not clear which part you are confused on.

But I'll try...

To define a curve as a function, we should be able, for each x value, to look along the vertical line at the x value, exactly one, and no more than one, value of y, to that x value. So at no x value, shall we look up and down the vertical line and find two or more y values. Notice that, for the circle, for x=0, we find two y values. Thus the circle cannot be defined as a function.

However, if we take the lower half, or the upper half, we can take just one of these halves, and we'll be able to describe it with a function.

Let us know if this does not clear things up, and precisely where you have a question.

 

[yaya10]

Thanks algebrat,

so, basically not every curve is a function.

so we say that x^2+y^2=1 is not a function because if we put x=0, we will have to solutions of y. y=sqrt(1). ...(1)


we can aslo say that y=x^2 is a function

but y^2=x is not a function for the same reason as (1)

Is that correct?


Thanks again for your help

 

[HallsofIvy]

Yes, that is true. It all reduces to the definition of "function". The very simplest definiton of "function" is "a set of ordered pairs such that no two distinct pairs have the same first member". That definition comes from the more primitive concept of a function where we start with some value, x (the "first member"), and produce a unique second value, y. I have always thought of this is being similar to the requirement in science that an experiment should be "reproducible"- that is, if do the same experiment repeatedly in exactly the same way (the x, or "first member"), we should get exact same results (the y, or second member). The definition is not "symmetric"- it is perfectly reasonable that doing an experiment in slightly different ways might give the same result. Another way of looking at it is that if you go into a store, you would find many different products at a variety of prices. It might well happen that very different products happen to have the same price. But if you found exactly the same product (same brand, same size, etc.) at different prices you would think that someone had made a mistake.

 

[yaya10]

Now, I see I think that's makes a lot of sense.

Like a Cola machine If you put $1 and press one button you will get one thing if you do it again you would have the same result.

I thank you both HallsofIvy and algebrat for your help:)


:)

Yaya