implicit function: Why implicit function is a function?

Why implicit function is a function?

Well, if it wasn't a function, it couldn't be called an implicit function ...

Can you say more about what you are thinking, maybe with an example?

The basic idea is that any equation with two variables defines a "relation"; that it, it determines what pairs (x,y) are "related" by satisfying the equation. In some cases, there is only one value of y that pairs up with any given value of x, and then the equation determines y as a function of x. For example, x + y = 5 determines a function, because for any value of x, you can solve for one value of y; but |x| + |y| = 5 does not determine a function, because, for example, if x = 3, y can be either 2 or -2. The former is an implicit function, because it doesn't explicitly tell you what y is; you have to figure that out. And some implicit functions can't actually be solved, so you can't write them explicitly at all, even though they are functions.

Beyond that, we'll need to see what examples you have, and why you think it might not be a function. It will also help if you tell us the definition you were given for "function", since some starting points make this easier to understand than others.
 
example of implicit function

i know that the circle equation is an implicit function.
The equation is: (x - a)^2 + (y - b)^2 = r^2
x, y = the variable
a, b = parameters
r = radius
(x,y) = point on the primeter of the circle.
So, why it is a function?
 
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i know that the circle equation is an implicit function.
The equation is: (x - a)^2 + (y - b)^2 = r^2
x, y = the variable
a, b = parameters
r = radius
(x,y) = point on the primeter of the circle.
So, why it is a function?
What is the definition of implicit function according to your book?

According to Wolfram (http://mathworld.wolfram.com/ImplicitFunction.html):

A function which is not defined explicitly, but rather is defined in terms of an algebraic relationship (which can not, in general, be "solved" for the function in question). For example, the eccentric anomaly
Inline1.gif
of a body orbiting on an ellipse with eccentricity
Inline2.gif
is defined implicitly in terms of the mean anomaly
Inline3.gif
by Kepler's equation

NumberedEquation1.gif


In your example, 'y' can be solved in terms of 'x'.

y = b \(\displaystyle \pm\sqrt{r^2 - (x-a)^2}\)

The above expression is sometimes defined as multi-valued function (it fails vertical line test)
 
i know that the circle equation is an implicit function.
The equation is: (x - a)^2 + (y - b)^2 = r^2
x, y = the variable
a, b = parameters
r = radius
(x,y) = point on the primeter of the circle.
So, why it is a function?

According to the usual definition of function, this is not a function, but a mere relation, as I discussed. If you were told that it is a function, can you quote what they said? They must be either using the term in a broad sense, or omitting some details. The context may be important. Yours is probably implicit differentiation in calculus, where we don't really care whether it is technically a function; a more accurate term would be "implicit curve".

For example, this page is careful about the distinction, and Wikipedia discusses it. Here is a nice explanation of the issues involved, pointing out that when we call such an equation an implicit function, we are really talking about only some piece of the curve, often the part near a given point, and with such a restriction it is a function.

Does this answer your issue?
 
I thought a circle is not a function because it fails the vertical line test?
For example, y=x^2 is a function but y= +- sqrt (x) is not.
 
I thought a circle is not a function because it fails the vertical line test?
For example, y=x^2 is a function but y= +- sqrt (x) is not.
As Dr. Peterson said, it all depends on how the word "function" is defined.

The standard modern definition is that a function maps each element of the set called the domain to exactly one element of the set called the range. So your understanding accords with the standard modern meaning.
 
I thought a circle is not a function because it fails the vertical line test?
For example, y=x^2 is a function but y= +- sqrt (x) is not.

Evidently I was right in guessing that this was your issue. But you haven't really answered our questions. How does your book define "implicit function"? What does it say in describing a circle as an implicit function? My guess would be that they are being somewhat informal and failing to explain the terminology, leading to your uncertainty.

Did you read the three links I included? I think they are all very helpful, especially the last.
 
Evidently I was right in guessing that this was your issue. But you haven't really answered our questions. How does your book define "implicit function"? What does it say in describing a circle as an implicit function? My guess would be that they are being somewhat informal and failing to explain the terminology, leading to your uncertainty.

Did you read the three links I included? I think they are all very helpful, especially the last.

Sorry i am not the OP? I was just putting in my pennies worth!!
 
I think a circle can be redefined as a function if expressed in parametric form. E.g
x= 2cost , y = 2sint is a parametric function that maps a parameter t to a pair of coordinates (x, y) . Sort of reals to realxreal? Would anyone agree?
But a circle expressed in implicit form doesn't feel like a function to me.
 
I think a circle can be redefined as a function if expressed in parametric form. E.g
x= 2cost , y = 2sint is a parametric function that maps a parameter t to a pair of coordinates (x, y) . Sort of reals to realxreal? Would anyone agree?
But a circle expressed in implicit form doesn't feel like a function to me.

Calling something a function is actually relative; you need to ask, "what is a function of what?" Every curve can (in principle) be expressed as a function of a parameter - you do that when you draw the graph, moving your pencil as a function of time. But that is an arbitrary introduction of another variable, not present in the equation itself. So you will notice (did you read those links?) that no one answers this question that way, because it doesn't help.

The issue is that in an implicit function we have two variables, and neither may be a function of the other, for the entire domain. Bringing in a third variable (parameter) changes the question. So the proper answer is that we describe it as a function in the sense that we can restrict the domain to make one variable a function of the other; and that restriction makes sense in any application of an implicit function, because we are paying attention only to the part near a given point.

But you are right that a parametric curve can be thought of as a function from R to R^2, namely f(t) = (x,y).
 
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