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## 6.14 Making Functions from Objects

There are many times when you compute a complicated expression and then wish to use that expression as the body of a function. Axiom provides an operation called function to do function:from an object this. function:made by function @{made by function} It creates a function object and places it into the workspace. There are several versions, depending on how many arguments the function has. The first argument to function is always the expression to be converted into the function body, and the second is always the name to be used for the function. For more information, see section MakeFunctionXmpPage .

Start with a simple example of a polynomial in three variables.

p := -x + y**2 - z**3

-z 3+y 2-x

Type: Polynomial Integer

To make this into a function of no arguments that simply returns the polynomial, use the two argument form of function.

function(p,'f0)

f0

Type: Symbol

To avoid possible conflicts (see below), it is a good idea to quote always this second argument.

f0

f0()==-z 3+y 2-x

Type: FunctionCalled f0

This is what you get when you evaluate the function.

f0()

-z 3+y 2-x

Type: Polynomial Integer

To make a function in x, use a version of function that takes three arguments. The last argument is the name of the variable to use as the parameter. Typically, this variable occurs in the expression and, like the function name, you should quote it to avoid possible confusion.

function(p,'f1,'x)

f1

Type: Symbol

This is what the new function looks like.

f1

f1 x==-z 3+y 2-x

Type: FunctionCalled f1

This is the value of f1 at x=3. Notice that the return type of the function is Polynomial (Integer), the same as p.

f1(3)
Compiling function f1 with type PositiveInteger -> Polynomial
Integer

-z 3+y 2-3

Type: Polynomial Integer

To use x and y as parameters, use the four argument form of function.

function(p,'f2,'x,'y)

f2

Type: Symbol
f2

f2(x,y)==-z 3+y 2-x

Type: FunctionCalled f2

Evaluate f2 at x=3 and y=0. The return type of f2 is still Polynomial(Integer) because the variable z is still present and not one of the parameters.

f2(3,0)

-z 3-3

Type: Polynomial Integer

Finally, use all three variables as parameters. There is no five argument form of function, so use the one with three arguments, the third argument being a list of the parameters.

function(p,'f3,['x,'y,'z])

f3

Type: Symbol

Evaluate this using the same values for x and y as above, but let z be -6. The result type of f3 is Integer.

f3

f3(x,y,z)==-z 3+y 2-x

Type: FunctionCalled f3
f3(3,0,-6)
Compiling function f3 with type (PositiveInteger,NonNegativeInteger,
Integer) -> Integer

213

Type: PositiveInteger

The four functions we have defined via p have been undeclared. To declare a function whose body is to be generated by function:declaring function, issue the declaration before the function is created.

g: (Integer, Integer) -> Float
Type: Void
D(sin(x-y)/cos(x+y),x)

-sin(y-x)sin(y+x)+cos(y-x)cos(y+x)cos(y+x)2

Type: Expression Integer
function(%,'g,'x,'y)

g

Type: Symbol
g

g(x,y)==-sin(y-x)sin(y+x)+cos(y-x)cos(y+x)cos(y+x)2

Type: FunctionCalled g

It is an error to use g without the quote in the penultimate expression since g had been declared but did not have a value. Similarly, since it is common to overuse variable names like x, y, and so on, you avoid problems if you always quote the variable names for function. In general, if x has a value and you use x without a quote in a call to function, then Axiom does not know what you are trying to do.

What kind of object is allowable as the first argument to function? Let's use the Browse facility of HyperDoc to find out. Browse@Browse At the main Browse menu, enter the string function and then click on Operations. The exposed operations called function all take an object whose type belongs to category ConvertibleTo InputForm. What domains are those? Go back to the main Browse menu, erase function, enter ConvertibleTo in the input area, and click on categories on the Constructors line. At the bottom of the page, enter InputForm in the input area following S =. Click on Cross Reference and then on Domains. The list you see contains over forty domains that belong to the category ConvertibleTo InputForm. Thus you can use function for Integer, Float, String, Complex, Expression, and so on.

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