3.5C Rational Functions and Asymptotes
A. Denition of a Rational Function
is said to be a rational function if
, where and are polynomial functions.
That is, rational functions are fractions with polynomials in the numerator and denominator.
B. Asymptote
3.5D Graphing Rational Functions
A. Strategy
1. Find all asymptotes (vertical, horizontal, oblique, curvilinear) and holes for the function.
2. Find the and intercepts.
3. Plot the and intercepts, draw the asymptotes, and plot enough points on the graph
t
2.7E Domain and Range of the Inverse Function
A.
Domain and Range of
1. To nd : nd
2. To nd
: nd
This comes from the idea that takes -values back to -values.
Warning: You have to use the above relationship. The output formula will give the wrong
domai
2.7D Inverse Functions II: Reections
A.
Introduction
It is sometimes undesirable to examine the inverse function by looking sideways. Thus, for
graphical purposes, we can get a non-sideways version of the graph of by switching
the and coordinates. Thus to
2.7F Capital Functions
A.
Motivation
If is useful, but not invertible (not one-to-one), we create an auxiliary function
similar to , but is invertible.
B.
that is
Capital Functions
Given , not invertible
, we dene
invertible.
must have the following prope
2.8 Distance Formula, Circles, Midpoint Formula
A. Distance Formula
We seek a formula for the distance between two points:
By the Pythagorean Theorem,
Since distance is positive, we have:
Distance Formula:
1
B. Example
Find the distance be
3.3A Review of Dividing Polynomials
A. Dividing Polynomials Using Algebraic Long Division
We use what is called algebraic long division.
You do the same steps as in long division of numbers, with a few extra things to consider.
Important Extra Features:
1