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Internet Resources for the Calculus Student


(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.

Composition of Functions

In addition to combining functions algebraically (eg by adding or multiplying their output values) we can also take the output of one and feed it in as the input to another (as for example to produce the square of sin(x) ). This process is called COMPOSITION of functions.

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.

Inverse Functions

In some cases two functions may have the property that one undoes the work of the other (like the cube and cube root for example). In such cases they are said to be inverse functions or composition inverses to one another. (Note that this is NOT the same as the algebraic inverse or reciprocal - eg the cube root is not one over the cube)

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 Yang.

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.

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