You may have seen the term 'polar functions' on various webpages in the context of graphing, but what does it actually mean? The simple answer is that it's a misnomer and in mathematics, there is no such thing as a 'polar function'. Functions are NOT categorized as "Cartesian" or "polar". The term polar is used to refer to coordinate systems and graphs, NOT functions.
The 'misnomer' arises from misunderstanding the terms polar function grapher or polar function plotter. When we use these terms, we are NOT referring to graphing or plotting 'polar functions'. We are simply referring to graphing or plotting functions using the polar coordinate system.
The word polar in these phrases modifies the word "grapher" and more precisely, it modifies "function grapher", not "function". Therefore, the more explicit term is function polar grapher or polar grapher of functions, both of which refer to a grapher that plots functions in the polar coordinate system. However, polar function grapher is the commonly used term. For more detailed explanations, please see Graphs of functions.
It should be noted that given any real valued function of real numbers, say y = f(x), can be graphed in either the Cartesian coordinate system or the polar coordinate system. For example, the graph of f(x) = x is a line in the Cartesian coordinate system and a spiral in the polar coordinate system (try graphing it using the function grapher in both coordinate systems).
Note that the choice of variables used to describe a function is a matter of convention. You can use θ and r instead of x and y, and write r = f(θ). This doesn't change anything; the graph of r = f(θ) = θ is the same line in the Cartesian coordinate systems and the same spiral in the polar coordinate system as they were for y = f(x) = x. You just need to rename the axes correspondingly. For example, in the Cartesian case you can call the independent axis as either the x-axis or θ-axis, and the dependent axis as either the y-axis or r-axis, depending on your choice of variables.
So, we know that any given function has both a Cartesian graph and a polar graph when plotted in their respective Cartesian and polar coordinate systems. However, it is a common misconception to categorize functions as either "Cartesian" or "polar". This categorization is erroneous and undermines the fundamental concept of a function. It can lead to confusion and difficulty in understanding the significance of graphically representing the same function in different coordinate systems.
To summarize, it is important to remember that the terms Cartesian and polar only refer to the type of graph or the coordinate system used to represent the function graphically, not the function itself. In other words, a function is not Cartesian or polar; it is simply a function. The term "polar functions" is a misnomer and should be avoided.