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### 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/
```
Associated Topics:
High School Equations, Graphs, Translations
High School Functions

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