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Last edited 11/14/2011
9:53:00 AM
FUNCTIONS:
IMAGES AND METAPHORS
A mathematician's mental representation of a function is generally quite rich and may involve many different metaphors and images kept in mind simultaneously.
Contents
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Math
object
Most
basic mental representation: A function is a mathematical object.
You must think of functions as math
objects when you are taking the rigorous view, which happens specificially when you are trying to prove something
about a function.
Example
This
example is intended to raise your consciousness about the possibilities for
functions as objects. (Read this, too.)
Consider the function defined by
.
(See its
graph.) Its value can be
computed at many different numbers but it is a single, static math object.
You can apply
operations to it
¨ Just as you can multiply a number by 2, you can multiply f by 2. You can say
“Let ”. Then
the value of g at any real x is
. You can add, subtract and multiply two real
functions as well. But you can’t do
arithmetic operations to functions that don’t have numerical input and output.
¨ . You can find its derivative. You can integrate it. (There are also plenty of functions you can’t differentiate or integrate.)
Like all math objects, functions may have properties.
¨ The function defined by is periodic with period π, meaning that for any x,
. It is
the function
itself that has period π, not any
value of it.
¨
Another property is that the graph
of f is bounded
by the horizontal lines y = 1
and y = 3.
As a math object, a function can be an element of a set.
¨ For example, f is an element of the set (MW, Wi) of real-valued functions that have derivatives of all orders. On
,
differentiation is an operator that takes a function in that set to another
function in the set. It takes f(x)
to
. Composition is a binary
operation on
(the chain rule shows that it preserves
differentiability.)
¨
If you restrict f to
the unit interval, it is an element of the function space . As such it is convenient to think of it as a point in the space (the whole function is the point, not just values of it). In this particular space, you can think of
the points as infinite-dimensional vectors and ask for the inner product of two
of them.
Expression
Mental representation: A function is an expression (or formula) to
evaluate.
The formula of the function f above allows you to compute its value at any x easily. But its formula is not the function. And not every function has a formula., for example the word length function or the nth prime
function. (There are programs that can
calculate those functions, but programs are not usually called expressions or
formulas.)
Function-as-expression is the image that motivates using the
defining
expression as the name of a function, as in “the derivative of is
” . This
works well for many functions studied in calculus.
The expression can tell you some properties of the function in a succinct way.
¨ You can tell by looking at the expression that is going to be a parabola with arms pointing
upwards (so it must have a minimum.)
¨ You can tell that is defined for all real x and that
is not.
¨ The expression ,
studied in the zoom and chunk chapter, requires a bit of work to
understand it, but, as I showed in that chapter, with work you can figure out
quite a lot about it.
¨ The expression for the elliptic function discussed here is an integral, so it is easy to see what its derivative is!
¨ Some expressions are so complicated it is hard to analyze them. The expression
is difficult because you can’t glance at it and see where its roots are. (The chapter on derivatives shows a graph of this function.) Nevertheless, it is a fifth degree polynomial, so you know for example that it is unbounded in both directions.
Don’t expect
that every function can be given by an expression
The word
length function and the nth prime
function cannot be given by expressions. The finite function can be given by the expression for
but the function is defined for only four
inputs (although of course the expression is defined for all real numbers)
so it is much easier
to see what it is like from the table.
It is useful to think of calculus-type functions as defined
by expressions
but don’t automatically look for expressions for a function
in general.
Process
Mental representation: A function is the process used to evaluate it at an input
Examples
¨
When you think of the function f defined by ,
your concern may be to know how to find its value at a given x. There
are various processes for doing this, for example:
a) You can do it in your head: Multiply x by 2, then add 2 to the result.
b) Since ,
you could add 1 to x, then multiply
the result by 2.
c) You can punch buttons on your calculator: Put in the value for x, multiply it by 2, then add 2 to the result. Or you could put in the value for x, add 1, then multiply the result by 2.
d) You could look at a graph of the function and measure the y value for the given x value.
e) You could write a program in some computer language that would compute the function.
There may be many ways to compute the value of a function.
The
computational process is not the same thing as the function.
|
In
the Olden Days, you would look up |
¨ If you were faced with the function you would probably punch buttons on your calculator,
since it is difficult to calculate sin x in
your head for most values of x. But for example if
,
you probably know that
,
so in that case you could do it in your head.
¨
¨ Even functions that don’t have formulas may still be computable using some process. The nth prime function can be computed at n by finding the first prime, finding the second prime, and so on until you get to n. Of course this is ridiculously tedious, but it is a process. You can also evaluate it on a computer, for example in Mathematica or Maple. These systems use a built-in process that is more efficient than finding one prime after another.
¨ A function defined by a table has a process for computing a value, too: Look it up in the table!
Edited to here
Graph
For a real function of one variable picturing its graph is
an extraordinarily useful help in understanding it. The
graph of (part
of which is shown at the right) suggests properties that it might have, such as
where its zeroes are. But its graph is not the function. And there are many functions whose graph you cannot
draw in any reasonable way.
Table of values
Dependency relation
Function as a dependency
relation. This is the metaphor behind such descriptions as “let x depend smoothly on t”. It is related to the graph point of view, but may not carry an
explicit picture; indeed, an explicit picture may be
impossible.
Transformer
A function does
something to its input. This point of view thinks of the function as a machine that transforms
the input; the function takes an object and turns it
into another object. In this picture, the function transforms
A variant of this metaphor is that
a function is a “black
box”, meaning that all that matters is input and output and
not how the input is changed into the output.
Algorithm
Function as algorithm or set of rules that tell you how to take an input and convert it into
an output. This is a metaphor related to
those of function as expression and as transformer, but the actual process is
implicit in the expression view and hidden in the transformer
(black box) view. Spell this out
Relocator
Function as relocator.
In this version, the function moves the point at
Map
Function as map is one of the most powerful metaphors in mathematics.
It takes the point of view that
the function renames the point labeled
.