In middle school, we have learned concepts such as the image of a function and how to draw it , and the image representation of a function .

　　Generally speaking, for functions

,

If the independent variable is taken as the abscissa and the dependent variable is taken as the ordinate, then a point is determined in the plane rectangular coordinate system . When traverses every value in the domain , the point set

It is called the graph of function  ( Figure 1 ) .

Figure 1： [ Animation ]

　　If the independent variable takes any value within the domain of definition, there is always only one corresponding function value. This kind of function is called a single-valued function. From a geometric point of view, that is: any straight line perpendicular to axis intersects the graph of the function at most at one point.

Figure 2： [ Animation ]

　　For example, the equation determines a function with as the independent variable and as the dependent variable on the closed interval . It is geometrically a circle with the center at the origin and the radius . It is easy to see that for each , there are two values corresponding to it (see Figure 3) , so is not a single-valued function. However, after additional conditions or , single-valued functions can be obtained respectively

or . [ Animation ]

But the form of the above circle equation in the polar coordinate system is

,

Therefore, it is obviously a single-valued function in  the polar coordinate system .

image 3

　　Note: From now on, unless otherwise stated, functions refer to single-valued functions.