|(including Composition, and Inverse Functions)|
One way of thinking of a function is as a rule or procedure for producing
a result which may depend on the value of some input. This could be like
a calculator button or formula, or an experimental procedure in which the
value of one quantity is set and the resulting value of another is taken
as the output. The important aspect of a function is that there is a unique
output for each possible input. It is possible to have variables related
in a way that does not determine one in terms of the other but such
relations are not functions.
A function is a special type of relation - one in which one of the variables is completely determined by the other.
When we name a function as say f, or g, or sin, or whatever, then we are referring to the relationship or operation connecting the two variables rather than the variables themselves. Thus a function is like a machine or calculator button which takes various possible values as input and gives the corresponding related value as output. The output from function f with input x is generally denoted by f(x) (which looks like, but must not be confused with, the notation often used for a product of two numbers).
Functions can arise from real-world situations, eg f(t) = position at time t of a moving object, or P(q)= profit to a manufacturer from producing q units of its product; or they may be given by formulas, eg f(t)=t^2 , (here we have used the "in-line" notation for a power both to simplify the formatting of this document and to familiarize you with how you may enter such things on a graphing calculator or computer algebra system).
The graph of a relationship between two real variables is the set of all points in a plane whose Cartesian coordinates x and y satisfy the relationship. For a function, f, this is the set of points (x,y) with y=f(x). A set of points that is the graph of a function must satisfy the "Vertical Line Test" - ie each vertical line can meet the graph in at most one point.
A nice introduction to the function concept is provided at the "CoolMath" site maintained by an instructor at a community college in California. And there are lots of other sites with similar material.
This graphing utility lets you see the graphs of functions specified by formulas that you can specify. This (or any graphing calculator or Computer Algebra System) can be used to help you understand and identify the symmetry properties of even and odd functions, and periodic functions, as well as properties such as asymptotes, and local maxima and minima.
Various operations on functions and their effects on the graphs are
illustrated in this
applet, and on this
page hosted at Vanderbilt University in the US. See also New
Functions from Old at Hofstra U. The effect of multiplying two
functions is illustrated by Karen's
Damping Functions Page at the "CoolMath" site, and a geometric construction
of the graph of 1/f(x)
is provided by the IES group in Japan.
An illustration of the composition of two functions is provided by this JAVA applet produced by the IES group in Japan.
The interactive DOS package composit.zip, from AZ-
MATH Software - University of Arizona, provides another interesting
demonstration of composition.
The concept of an inverse function is also explained at Inverse Functions, from California State University San Bernardino's Reference Notes Page.
Also see One-to-One
Functions and Inverse Functions, and How
to find the inverse? from Wei-Chi
And there are lots of other sites with similar material.
If you have come across any other good web-based illustrations of these
and related concepts, please do let
us know and we will add them here.
|Review Contents of Math&Stats Dep't Website||.......||Give Feedback||.......||Return to Langara College Homepage|