Is it possible to define this explicitly?

  • Thread starter Nano-Passion
  • Start date
In summary: I think it might be helpful to think of this in terms of a graph. If you were to plot the function, it would look something like this:In summary, the function doesn't have a unique y value for every x. It has two values that depend on the x value.
  • #1
Nano-Passion
1,291
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(x-3)^2 + (y-2)^2 = 1

With some rearranging I get

y(y-4) = -x^2+6x-13

Is it possible to define this explicitly in terms of y? It looks like a mathematical impossibility to define it explicitly in terms of y but I am hoping there might be some analytical technique for this.
 
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  • #2
What does "define explicitly" even mean? The formula at the start explicitly defines a circle.
Do you want a function ##y(x)##? Then it should be easy to see the answer is no: for every x there are two values of y.
 
  • #3
pwsnafu said:
What does "define explicitly" even mean? The formula at the start explicitly defines a circle.
Do you want a function ##y(x)##? Then it should be easy to see the answer is no: for every x there are two values of y.

Define the function explicitly in terms of y.

And okay.
 
  • #4
I don't see why not being single-valued means it's impossible to define something explicitly.

##(y-2)^2 = \text{something}##
##(y-2) = \pm \sqrt{\text{something}}##
...
 
  • #5
AlephZero said:
I don't see why not being single-valued means it's impossible to define something explicitly.

##(y-2)^2 = \text{something}##
##(y-2) = \pm \sqrt{\text{something}}##
...

Well in this case it is more like (y^2-4y) = something. so it is bit more complicated. Who knows, maybe one day some genius might develop a clever theory to solve it.
 
  • #6
AlephZero said:
I don't see why not being single-valued means it's impossible to define something explicitly.

##(y-2)^2 = \text{something}##
##(y-2) = \pm \sqrt{\text{something}}##
...

This is why I asked what "define explicitly" means.

Nano-Passion said:
Well in this case it is more like (y^2-4y) = something. so it is bit more complicated. Who knows, maybe one day some genius might develop a clever theory to solve it.

:confused: How did you get that?
Edit: And why not just complete the square?
 
  • #7
Nano-Passion said:
(x-3)^2 + (y-2)^2 = 1

With some rearranging I get

y(y-4) = -x^2+6x-13

Is it possible to define this explicitly in terms of y?
You can solve the first equation for y in terms of x. However, the equation defines a relation between x and y, but doesn't represent a function. Some values of x map to two different y values.

Having said all that, it's pretty simple to solve for y. Leave the terms in y on one side, and move the terms in x to the other side. Do not expand the (y - 2)2 term. Then take the square root of both sides, remembering to use ± for the two solutions.
Nano-Passion said:
It looks like a mathematical impossibility to define it explicitly in terms of y but I am hoping there might be some analytical technique for this.
 
  • #8
Mark44 said:
You can solve the first equation for y in terms of x. However, the equation defines a relation between x and y, but doesn't represent a function. Some values of x map to two different y values.

Having said all that, it's pretty simple to solve for y. Leave the terms in y on one side, and move the terms in x to the other side. Do not expand the (y - 2)2 term. Then take the square root of both sides, remembering to use ± for the two solutions.

Oh, now I see what Alephzero was saying.
 

1. Is it possible to define something explicitly in science?

Yes, it is possible to define something explicitly in science. In fact, defining concepts and terms explicitly is a crucial aspect of the scientific method. By defining something explicitly, scientists can ensure that everyone is using the same understanding and meaning of a term or concept, which allows for more accurate and replicable research.

2. How do scientists define concepts and terms explicitly?

Scientists define concepts and terms explicitly by using precise and specific language, providing clear and detailed descriptions, and referencing established definitions or theories. This ensures that there is a standard understanding and usage of the defined concept or term within the scientific community.

3. Why is it important to define things explicitly in science?

Defining things explicitly in science is important because it allows for clear communication and understanding among scientists. It also ensures that research and experiments are replicable, as everyone is using the same definitions and concepts. Additionally, explicit definitions help to avoid confusion and potential errors in data interpretation.

4. Are there any limitations to defining things explicitly in science?

While defining things explicitly is important in science, there are some limitations. Some concepts and terms may be difficult to define explicitly, as they can be complex and multifaceted. Also, language and definitions can vary between different fields and disciplines in science, making it challenging to have a universal definition for certain concepts.

5. Can explicit definitions change in science?

Yes, explicit definitions can change in science. As new research and discoveries are made, definitions may need to be updated or revised to reflect a better understanding of a concept or term. This is why it is important for scientists to constantly review and refine their definitions to ensure accuracy and advancement in the field of science.

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