Domain, Range, and Asymptote

Date: 10/12/98 at 21:29:26
From: Summer
Subject: Pre-Calculus

How do you find the domain, range, and asymptote of a function? Please 
help.

Date: 10/13/98 at 11:56:25
From: Doctor Rob
Subject: Re: Pre-Calculus

The domain of a function is the set of values you can substitute for 
the variable and get a sensible answer. Usually it is a subset of some
implicitly agreed-upon set, such as the real numbers, or the complex
numbers, or the integers. Nonsensical answers include those which cause 
division by zero, square roots of negative numbers (in the real number 
case), arcsines of numbers larger than 1, logarithms of negative 
numbers (in the real number case), and so on.

The range of a function is the set of values you get as results when 
you substitute the values in the domain for the variable.

There are three kinds of asymptotes: vertical, horizontal, and oblique.

Vertical asymptotes can be found by finding the values of x where the
function grows without bound nearby. Often they are values of x where 
the denominator vanishes, but not always. Example: f(x) = 3/(x-1) has 
a vertical asymptote at x = 1. Example: f(x) = log(x) has a vertical
asymptote at x = 0.

Horizontal asymptotes are constant values that f(x) approaches as x 
grows without bound. Example: f(x) = 3/(x-1) has a horizontal asymptote 
at f(x) = 0.

Oblique asymptotes are first degree polynomials which f(x) gets close 
as x grows without bound. Example: f(x) = (2*x^2+3*x+1)/(4*x-1) 
approaches the first-degree polynomial g(x) = (1/2)*x + 7/8 as x grows 
without bound.

In all cases, asymptotes represent straight lines in the plane, either
x = c or y = g(x), which are approached by the graph of y = f(x) as y 
or x or both grow without bound.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/