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College Math Resources - Topics in Precalculus

 Functions 

 (including  Composition, and Inverse Functions) other sites

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.

There are lots of other sites on the web where you can find more information and examples on this topic.

 

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.
  Check for Links on Composite Functions


 

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).
  Check for Links on Inverse Functions




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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