Objectives: In this tutorial, we define the inverse of a function. We discuss three different ways of determining whether a function has an inverse. Some examples are discussed. After working through these materials, the student should be able

• to recognize from the graph of a function whether the function has an inverse;
• to recognize from the graph of a function whether the function is one to one;
• to graph the inverse of a function;
• to algebraically find the inverse of a function;
• to algebraically show that a function is not one to one.
Modules:
 Definition. Let f be a function with domain D and range R. A function g with domain R and range D is an inverse function for f if, for all x in D, y = f(x) if and only if x = g(y).

• Discussion [Using Flash]

• From this discussion, we have the following.

 Definition. A function f is one to one if, given two distinct points a and b in the domain of f, the images f(a) and f(b) are different points. Equivalently, A function f is one to one if, given two points a and b in the domain of f such that f(a) = f(b), then a = b.

 Theorem. Let f be a function with domain D and range R. The following are equivalent: f has an inverse g. There is a function g such that gf(x) = x for all x in D and fg(y) = y for all y in R. f is one to one. Horizontal Line Test. Each horizontal line meets the graph of f in at most one point.

 Corollary. If the function f is increasing or decreasing then f has an inverse g. In addition, f is increasing if and only if its inverse g is increasing. f is decreasing if and only if its inverse g is decreasing.

• Discussion

 Notation. The inverse function for f is denoted by f -1.

• Examples:

• Drill in finding the inverses of functions.