Domain, Range, and AsymptoteDate: 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/ |
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