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The y value corresponding to x = 3 is y = f ( 3) = 11/2 Therefore the point ( 3, 11/2) lies on the graph of f Similarly, f ( 4) = 6 so that ( 4, 6) lies on the graph However, f is undefined at x = 1, so there is no point on the graph with x coordinate 1 The sketch in Figure 138 was obtained by plotting several points

The type parameter T is specified by ByTwos and is also specified in ISeries This is important A class that implements a generic version of a generic interface must, itself, be generic For example, the following declaration would be illegal because T is not defined:

class ByTwos : ISeries<T> { // Wrong!

The type argument required by the ISeries interface must be passed to ByTwos Otherwise, there is no way for the interface to receive the type argument

Notice that, for each x in the domain of the function, there is one and only one point on the graph---namely the unique point with y value equal to f (x) If x is not in the domain of f, then there is no point on the graph that corresponds to x

Part I:

Next, the current value of the series, val, and the starting value, start, are declared to be objects of the generic type T Then, a delegate called IncByTwo is declared This delegate defines the form of a method that will be used to increase an object of type T by two In order for ByTwos to work with any type of data, there must be some way to define what an increase by two means for each type of data This is achieved by passing to the ByTwos constructor a reference to a method that performs an increase by two This reference is stored in incr When the next element in the series is needed, that method is called through the incr delegate to obtain the next value in the series Notice the class ThreeD It encapsulates three-dimensional (X,Z,Y) coordinates It is used to demonstrate ByTwos on a class type In GenIntfDemo, three increment methods are declared; one for int, one for double, and one for objects of type ThreeD These are passed to the ByTwos constructor when objects of their respective types are created Pay special attention to ThreeDPlusTwo( ), shown here:

// Define plus two for ThreeD static ThreeD ThreeDPlusTwo(ThreeD v) { if(v==null) return new ThreeD(0, 0, 0); else return new ThreeD(vx + 2, vy + 2, vz + 2); }

Observe that, corresponding to x = 3, for instance, there are two y values on the curve Therefore the curve cannot be the graph of a function

Notice that it first checks if v is null If it is, then it returns a new ThreeD object in which all fields are set to zero The reason for this is that v is set to default(T) by the ByTwos constructor This value is zero for value types and null for object types Thus, (unless SetStart( ) has been called) for the first increment, v will contain null instead of a reference to an object This means that for the first increment, a new object is required A type parameter for a generic interface can have constraints in the same way as it can for a generic class For example, this version of ISeries restricts its use to reference types: