Rational function: Difference between revisions
imported>Mirzhan Irkegulov (New page: {{subpages}} '''Rational function''' is a quotient of two polynomial functions. It distinguishes from ''irrational function'' which cannot be written as a ratio of two [[polyno...) |
imported>Daniel Mietchen |
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where ''s'' and ''t'' are [[polynomial function]] in ''x'' and ''t'' is not the [[zero polynomial]]. The [[domain (mathematics)|domain]] of ''f'' is the set of all points ''x'' for which the denominator ''t''(''x'') is not zero. | where ''s'' and ''t'' are [[polynomial function]] in ''x'' and ''t'' is not the [[zero polynomial]]. The [[domain (mathematics)|domain]] of ''f'' is the set of all points ''x'' for which the denominator ''t''(''x'') is not zero. | ||
On the graph restricted values of an axis | On the graph restricted values of an axis form a straight line, called [[asymptote]], which cannot be crossed by the function. If zeros of [[numerator]] and [[denominator]] are equal, then the function is a horizontal line with the hole on a restricted value of ''x''. | ||
==Examples== | ==Examples== | ||
Let's see an example of <math>f(x) = \frac{x^2-x-6x}{x^2+x-20}</math> in a factored form: <math>f(x) = \frac{(x+2)(x-3)}{(x+5)(x-4)}</math>. Obviously, roots of denominator is -5 and 4. That is, if ''x'' takes one of these two values, the denominator becomes equal to zero. Since the division by zero is impossible, the function is not defined or discontinuous at ''x'' = -5 and ''x'' = 4. | Let's see an example of <math>f(x) = \frac{x^2-x-6x}{x^2+x-20}</math> in a factored form: <math>f(x) = \frac{(x+2)(x-3)}{(x+5)(x-4)}</math>. Obviously, roots of denominator is -5 and 4. That is, if ''x'' takes one of these two values, the denominator becomes equal to zero. Since the [[division by zero]] is impossible, the function is not defined or discontinuous at ''x'' = -5 and ''x'' = 4. | ||
The function is continuous at all other values for ''x''. The domain (area of acceptable values) for the function, as expressed in interval notation, is: <math> (-\infty; -5) \cup (-5; 4) \cup (4; \infty) </math> | The function is continuous at all other values for ''x''. The domain (area of acceptable values) for the function, as expressed in [[interval notation]], is: <math> (-\infty; -5) \cup (-5; 4) \cup (4; \infty) </math> | ||
Latest revision as of 05:28, 9 December 2008
Rational function is a quotient of two polynomial functions. It distinguishes from irrational function which cannot be written as a ratio of two polynomials.
Definition
A rational function is a function of the form
where s and t are polynomial function in x and t is not the zero polynomial. The domain of f is the set of all points x for which the denominator t(x) is not zero.
On the graph restricted values of an axis form a straight line, called asymptote, which cannot be crossed by the function. If zeros of numerator and denominator are equal, then the function is a horizontal line with the hole on a restricted value of x.
Examples
Let's see an example of in a factored form: . Obviously, roots of denominator is -5 and 4. That is, if x takes one of these two values, the denominator becomes equal to zero. Since the division by zero is impossible, the function is not defined or discontinuous at x = -5 and x = 4.
The function is continuous at all other values for x. The domain (area of acceptable values) for the function, as expressed in interval notation, is:
