Topics for this Lab:	Standard Maple Functions	Composition of Functions	Piecewise Functions	Special Points	Unapply

Expressions are a useful tool, however they can become confusing as the variables all are global. Changing x changes the value of any expression involving x. This is not usually how we want things to behave.

We really want to be able to define a mathematical function and use the f(x) notation. To do this we use the -> operator.
This tells Maple to define the function f(x) = x^2+3*x+4, the global value of x is irrelevent. Try
> f := x -> x^2+3*x+4;
> x:=7;
> f(2);
> f(n);

1. Define f(x) to be the polynomial x^3-2x^2-3x+2 and g as the function n^2.
   1. Find f(2), f(56), f(-28) and f(304).
   2. Find f as a function of y, i.e. f(y).
   3. Find f(g(x)).
   4. Find g(f(x)).

Standard Maple Functions

Calculus Functions:
exp	the exponential function: exp(x) = sum(x^i/i!,i=0..infinity) = x -> E^x
ln	natural logarithm (logarithm with base E = 2.71828...)
log	logarithm to arbitrary base
log10	log to the base 10
max, min	maximum/minimum of a list of real values
sqrt	square root

The trigonometric functions:		sin, cos, tan, sec, csc, cot.
The hyperbolic functions:		sinh, cosh, tanh, sech, csch, coth
The inverse trigonometric functions:		arcsin, arccos, arctan, arcsec, arccsc, arccot, arctan.
The inverse hyperbolic functions:		arcsinh, arccosh, arctanh, arcsech, arccsch, arccoth.

Complex Functions:
argument	argument of a complex number or expression
conjugate	conjugate of a complex number or expression
abs	absolute value of real or complex argument

Integer functions:
binomial	binomial coefficients: binomial(n,r) = n!/(r!*(n-r)!)
ceil	ceil(x) = smallest integer greater than or equal to x
factorial	the factorial function factorial(n) = n!
floor	floor(x) = greatest integer less than or equal to x
round	round(x) = nearest integer to x (round(.5) = 1)
signum	sign function for real and complex expressions
trunc	trunc(x) = nearest integer from x in the direction of 0

You can use Maple's help to find the exact syntax of these functions (though most of them are obvious). To use Maple's help type in the word and press control F1, or use the keyword search or browser from the help menu.

2. Define f(x) to be the function sin(x)/x
   1. Find f(y).
   2. Find f(0.005).
   3. Find f(0.000000000001).
   4. Set Digits to 100 and find f(0.000000000001).
   5. Try to find f(0) - what happens and why?
   
   
3. Define f(x) to be the function ln(x), g(x) to be sqrt(x) and h(x) to be f(x)/g(x).
   1. Find f(-1).
   2. Find g(-1)
   3. Find h(-1).
   4. Why is the answer for h incorrect if we are working over the reals?

The last question shows that we must be careful about function domains in Maple.

Composition of Functions

Given two functions f and g, we may define f composed with g by using the @ symbol.

> f:=x->sqrt(x);
> g:=x->sin(x);
> (f@g)(x);
Note the brackets around f@g.

4. Define f(x) to be the function x², g(x) to be sin(x) and h(x) to be cos(x).
   1. Find f composed with g.
   2. Find f composed with h.
   3. Find the sum of f composed with g and f composed with h.

Piecewise Functions

NOTE: piecewise only works in version 4 or above of Maple. In version 3 you must use the Heaviside function to define piecewise functions.

In order to define ranges you may use any of the following symbols:

5. Define the following functions:
   

   1. f(x)	= {	ex	if x < 0
      		sin2(x)	if 0 <= x < PI
      		sqrt(cos(x))	Otherwise

   2. The Federal tax rate is 0% for income under $6000, 17% for income from $6000 to $25,000, and 26% for income over $30,000. Write a function which calculates the tax payable given the income.
   3. Provincial tax is 17% of Federal tax. Write a function to compute total tax payable given the federal tax payable.
   4. Use composition to create a function which computes the total tax payable (Federal and Provincial), given the income.

Special points

Certain functions have special values defined at particular points. For example

f(x)	= {	1	if x = 0
		x/x	otherwise

To do this in Maple we redefine these points after defining the function. Thus the above function would be
>f(x):=x->x/x;
>f(0) := 1;

We can define as many points as we like in this way.

Changing Between Functions and Expressions

unapply turns an expression into a function.
The syntax is unapply(a, x), where a is an expression and x is the variable in the definition of a which is to be made into a free variable.
> x:='x';
> a:=x^2;
> f:=unapply(a,x);
> a;
> x:=2;
> a;
> f(y);
> f(4);

Note that x must be undefined when unapply is executed. The following will generate an error.
> a:=x^2;
> x:=2;
> f:=unapply(a,x);
So will
> a:=x^2;
> x:=c+1
> f:=unapply(a,x);

However the following is OK.
> a:=x^2;
> x:=c+1
> f:=unapply(a,c);

6. Turn the following Expressions into functions f, g, h etc, the variables on the right are always assumed to be free.
   1. a:=x^2+1;
   2. y:=sin(a);
   3. x:=ln(y);

It is always possible to turn a function into an expression by assigning it. That is, if f is a function then f(x) is an expression.
> f:=x->x^2;
> a:=f(z);
Now try
> a;
> z:=2;
> a;
> f(5);

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