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`.

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`.

Functions can be used without assigning them to variables.

Another way to make new functions is to compose two old ones with the operator @@.

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.

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`.

i1 : sq = i -> i^2 o1 = sq o1 : FunctionClosure |

i2 : sq 10 o2 = 100 |

i3 : sq(5+5) o3 = 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 |

i6 : (i -> i^2) 7 o6 = 49 |

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) |

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) |

The class of all functions is Function.