According to the definition of implicit function we cannot determine the value of one variable explicitly from the function. I have gone through many websites, few books and few youtube videos but could not understand implicit function clearly.
Equation of a circle is an implicit function as mentioned in most of the sources because we cannot get the value of y from the function f(x,y)=x^2+y^2=25 explicitly.But why can't we get the value of y explicitly i mean according to this function we can get two values for y at a certain abcissa. We are clearly getting two values of y for a certain value x then why is it implicit function?
I could not get the concept of this type of function. Pardon me for writing anything wrong if I have done any.Looking for an explanation
Again I apologise for asking such a silly question in this forum.
One can find a rigorous definition in Wikipedia , however I will attempt to provide a practical way of thinking of implicit function's.
An explicit single variable function can be thought of as a 'mapping' from one set to another set of the form $x \to y$, this can be graphed on a cartesian grid by highlighting all the pair $(x,y)$ included in this mapping.
Now, in an implicit function, we no longer have the idea of mapping from the previous definition but we can think of the solution's to the implicit equation which defines the function. If we took a graph and highlighted all the points which solved the equation of the implicit function, then we would get the curve of the implicit function analogous to how there was a curve corresponding to the explicit function.
Sometimes we can write implicit functions as a 'multiple' explicit function, for example we see that when writing the circle's equation:
$$ y = \begin{cases} \sqrt{1-x^2} \\ - \sqrt{1-x^2} \end{cases}$$
Notice that we have two possible definitions of 'y' when we isolate the implicit equation and solve for $y$, this two definitions can be thought of as pieces of the original curve. For example, $y=\sqrt{1-x^2}$ defines the upper disc and $ y = - \sqrt{1-x^2}$ defines the lower one.