Rational function: Difference between revisions
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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>[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 06:00, 10 October 2024
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:
