1. Definitions
   • function

     a rule that assigns to each element of one set (the domain) exactly one element of another set (the range).

     Functions can be defined in various ways: by

     • a formula
     • data
     • a graph
     • a verbal description

     Types of functions:

     • linear functions (whose graphs are lines)
     • non-linear functions (whose graphs are not straight lines)
     • step functions
     • constant functions
   • domain of a function

     is taken to be the largest set consistent with the definition of the function and the specification of the problem

   • step function

     a function whose graph looks like a staircase (in particular it has ``jumps'' between the steps: this type of function is called discontinuous)

2. Theorems/Formulas

3. Properties/Tricks/Hints/Etc.

   vertical line test: if a vertical line intersects a graph in more than one point, then the graph is not the graph of a function. A violation corresponds to an element of the domain with two different corresponding range elements, which must not occur.

   compositions: given functions f(x) and g(x). If you see the expression f(g(x)), it means that you should first evaluate y=g(x), and then evaluate f(y).

   When you're asked to find the domain of a function, given a formula, you should give the largest set possible that makes sense. In particular, you should look out for square roots (their arguments should be positive) and for quotients (the demoninators should never be zero).

4. Summary

   This section provides a formal definition of a function, gives examples (various types and specific examples), and discusses how to graph them, how to find domains, and gives a little about compositions of functions.