The operator
> is used to make new functions. On its left we provide the names of the parameters to the function, and to the right we provide the body of the function, an expression involving those parameters whose value is to be computed when the function is applied. Let's illustrate this by making a function for squaring numbers and calling it
sq.
i1 : sq = i > i^2
o1 = sq
o1 : FunctionClosure

i2 : sq 10
o2 = 100

i3 : sq(5+5)
o3 = 100

When the function is evaluated, the argument is evaluated and assigned temporarily as the value of the parameter
i. In the example above,
i was assigned the value
10, and then the body of the function was evaluated, yielding
100.
Here is how we make a function with more than one argument.
i4 : tm = (i,j) > i*j
o4 = tm
o4 : FunctionClosure

i5 : tm(5,7)
o5 = 35

Functions can be used without assigning them to variables.
i6 : (i > i^2) 7
o6 = 49

Another way to make new functions is to compose two old ones with the operator
@@.
i7 : sincos = sin @@ cos
o7 = sincos
o7 : FunctionClosure

i8 : sincos 2.2
o8 = .555114915759425
o8 : RR (of precision 53)

i9 : sin(cos(2.2))
o9 = .555114915759425
o9 : RR (of precision 53)

Code that implements composition of functions is easy to write, because functions can create new functions and return them. We illustrate this by writing a function called
comp that will compose two functions, just as the operator
@@ did above.
i10 : comp = (f,g) > x > f(g x)
o10 = comp
o10 : FunctionClosure

i11 : sincos = comp(sin,cos)
o11 = sincos
o11 : FunctionClosure

i12 : cossin = comp(cos,sin)
o12 = cossin
o12 : FunctionClosure

i13 : sincos 2.2
o13 = .555114915759425
o13 : RR (of precision 53)

i14 : cossin 2.2
o14 = .690586688560911
o14 : RR (of precision 53)

We created two composite functions in the example above to illustrate an important point. The parameters
f and
g acquire values when
sincos is created, and they acquire different values when
cossin is created. These two sets of values do not interfere with each other, and the memory they occupy will be retained as long as they are needed. Indeed, the body of both functions is
x > f(g(x)), and the only difference between them is the values assigned to the parameters
f and
g.
The class of all functions is
Function.