Inverse Functions

Tom Kelliher, MA 114

Oct. 27, 1998

Announcements

Collect homework. Questions on homework? Exam next Friday. Trig.\ situation.

From Last Time

1. Operations on functions, word problems

Outline

1. Inverse function, theorem, graphing.

Assignment

Read 3.1. Problems due Monday from 2.7: 4, 6, 16, 26, 36, 42.

Inverse Functions

Definitions:

1. A function is one-to-one if two values from the domain never map to the same value in the range. Algebraically: if , then .

2. Horizontal line test: If a function is one-to-one, no horizontal line intersects it at more than one point. (Be careful when graphing. Consider . Zoom or trace to make sure.) If a function passes the horizontal line test, then it is one-to-one.

3. Let f be a one-to-one function. Then f has an inverse function, g, whose domain is the range of f and whose range is the domain of f. g's rule is: exactly when .

   

   Do and have inverses? If so, find them algebraically.

Inverse Theorem

Suppose f and g are functions such that

Then f is one-to-one and its inverse is g.

Show that is the inverse of .

Graphing Inverse Functions

1. Does have an inverse?

2. How do find it?

3. What's the parametric representation of ? What do we have if we swap the two equations?

4. Observe that is on the graph of f exactly when b,a is on the graph of f's inverse.

5. If g is the inverse of f, then g's graph is f's graph reflected about the axis y = x.

Often, the inverse of is denoted . This is not division.

Word Problems

If there's time, look at some of the word problems on pg. 147.