Thread: [SOLVED] Implicit functions and relevant questions.

I am plagued with these doubts. Can any of you dispel them?

Question 1. Say, we have a curve in the x-y plane. This curve is continuous in a certain domain but haphazard (arbitrary) with/without the possibility of double or more values. Can this line represent some function y=f(x)? Note that we don't start with a given function y=f(x) and from there go on to construct the curve. It is the other way round: we have some arbitrary curve and we want to know if a corresponding elementary function exists. In other words we want to explore the existence of the elementary function corresponding to the arbitrary curve. Or is there a possibility that such a curve can represent a higher transcendental function which cannot be expressed in the form of y=f(x), but can be given only in series form? [A digression: this brings me to the question whether higher transcendental functions can at all be expressed as curves on the x-y plane?]

Question 2. Existence Theorem of implicit function, F(x,y)=0 : it has two meanings for me. Which of them is correct (if at all)?
On one hand, it seems to mean that an implicit function will not exist if there are no points (x,y) that satisfy F(x,y)=0. Geometrically speaking, we can visualize the surface given by F(x,y) as not touching/cutting the x-y plane anywhere. So for implicit functions with more variables, but there the visualization is not possible as we are tied up to a 3D world.
On the other hand, existence can mean that there are points satisfying F(x,y)=0 but such an array of points is not capable of defining a function y=f(x). If this is so then it takes us back to Question 1 that I have posed.
Which (if any) is true?

Question 3. Suppose we have a function $\displaystyle y=x^2$. We can define this as a function where x is the argument. As x changes over a given domain of definition, y takes up corresponding values. We may also interpret it as meaning $\displaystyle x=\sqrt{y}$. Then the independent variable is y and the dependent variable is x. But that is by prior agreement as to how we define the function. Though in either case the curve traced out in the x-y plane is one and the same (after rearranging the axes when plotting).
Next, if we have a function $\displaystyle z=x^2 + xy + y^2$, where the variables are not separable, my question is, can we say, in the same vein as before, that x is a function of the independent variables y and z, or again, that it is equally true that y is a function of the independent variables x and z, though originally the function was defined with z as dependent and x and y as independent variables? This kind of argument is unnecessary since the dependent and independent variables are always defined beforehand. But, to my mind, such an ambiguous situation crops up as soon as a function becomes implicit. With F(x,y)=0, one dimension, the z, is gone and we are down to the x-y plane with an arbitrary curve showing the place where F(x,y) had cut the x-y plane, viz. the array of points for which F(x,y)=0.
This brings me to my fourth question.

Question 4.Suppose there is an implicit function F(x,y,z)=0. Then which one is true: z is a function of x and y? or
y is a function of x and z? or
x is a function of y and z?

Thank you.
Regards,
fanofandrew

Originally Posted by fanofandrew

Question 3. Suppose we have a function $\displaystyle y=x^2$. We can define this as a function where x is the argument. As x changes over a given domain of definition, y takes up corresponding values. We may also interpret it as meaning $\displaystyle x=\sqrt{y}$. Then the independent variable is y and the dependent variable is x. But that is by prior agreement as to how we define the function. Though in either case the curve traced out in the x-y plane is one and the same (after rearranging the axes when plotting).

The curves represented by $\displaystyle f(x,y)=y-x^2=0$ and by $\displaystyle g(x,y)=x-\sqrt{y}=0$ are not the same. The first is the parabola $\displaystyle y=x^2, x \in \mathbb{R}, $ while the second is the half parabola $\displaystyle y=x^2, x \in \mathbb{R_+}.$ This is because $\displaystyle \sqrt{}$ of a positive number is defined to be positive by convention.

RonL

Originally Posted by fanofandrew

Question 1. Say, we have a curve in the x-y plane. This curve is continuous in a certain domain but haphazard (arbitrary) with/without the possibility of double or more values. Can this line represent some function y=f(x)? Note that we don't start with a given function y=f(x) and from there go on to construct the curve. It is the other way round: we have some arbitrary curve and we want to know if a corresponding elementary function exists. In other words we want to explore the existence of the elementary function corresponding to the arbitrary curve. Or is there a possibility that such a curve can represent a higher transcendental function which cannot be expressed in the form of y=f(x), but can be given only in series form? [A digression: this brings me to the question whether higher transcendental functions can at all be expressed as curves on the x-y plane?]

Let $\displaystyle C$ denote a curve on some domain $\displaystyle D$, and suppose that $\displaystyle C$ describes an unknown function then of course you can create a function that graphs the curve. Suppose that there is no elementary or transcendental function that describes the function $\displaystyle C$ over the entire domain $\displaystyle D$, then we may subdivide $\displaystyle D$ into as many subdomains as we wish and create a piece-wise function that describes the curve on each of the subdomains respectively.

Originally Posted by fanofandrew

Question 4.Suppose there is an implicit function F(x,y,z)=0. Then which one is true: z is a function of x and y? or
y is a function of x and z? or
x is a function of y and z?

Are you saying that if we have some function $\displaystyle F(x,y,z,\cdots)=0$ how can we discern which is the independent and dependent variables? I do not think that can be done, but as your question is stated the answer is all of the above. First let us look at $\displaystyle y=f(x)$, we have that y is a function of x, but conversely supposing that $\displaystyle f(x)$ is bijective on some domain then on that domain we may say that $\displaystyle x=f^{-1}(y)$ in which case we have that x is a function of y, but the functions are the same.

Thanks a lot. The grey area in my mind has cleared with your replies. I conclude that when a certain implicit function f(x,y)=0 does not exist it simply means that f(x,y) does not cut or touch the x-y plane.
Regards,
fanofandrew