I think what you are asking is: "If I have a curve on a plane,how do I know if it is a function or not?". For regular Cartesiancoordinates, where x is the independent variable and y is thedependent variable, you can use something called the "vertical linetest". If you hold up a line parallel to the y-axis it willintersect the curve. Now if you can move it side to side and itnever intersects the curve *more than once* then you have afunction (of x).The reason is this: a function is a rule that associates everypoint in the domain to a single point in the range (also called thecodomain). This means that if you are given any point in the domainand evaluate the function at that point you will get one value inthe range (this does not need to be a unique value. You can havetwo different points in the domain taking on the same value in therange; think absolute value or sine curves). So, if we are usingthe vertical line test and it intersects the curve twice for asingle value of x, we know that the curve cannot be a function,since there are two values in the range associated with one valuein the domain. On the other hand, if the curve passes the verticalline test i.e. it only intersects the curve once at every point inthe domain, then you have a function of x.You can use an analogous "horizontal line test" to see ifsomething is a function of y.