Rational Functions: Definition and Properties

This page introduces the concept of rational functions, their canonical form, and key characteristics. Funkcja wymierna wzory are essential for understanding these mathematical entities.

A rational function is defined as a function of the form f$x$ = P$x$ / Q$x$, where P$x$ and Q$x$ are polynomials, and Q$x$ ≠ 0. The page provides examples of rational functions, such as f$x$ = 3/x and g$x$ = $x^2 + 2x - 1$ / $x^2 + x - 3$.

Definition: A rational function is a function expressed as the ratio of two polynomials, where the denominator is not equal to zero.

The canonical form of a rational function is presented as f$x$ = a / $x - p$ + q, which is useful for analyzing the function's behavior and graphing.

Example: The function f$x$ = 1 / $x + 2$ + 5 is in canonical form, where a = 1, p = -2, and q = 5.

The page also introduces the concept of asymptotes, which are important features of rational function graphs. Vertical asymptotes occur where the denominator equals zero, while horizontal asymptotes represent the function's behavior as x approaches infinity.

Highlight: Understanding asymptotes is crucial for sketching wykres funkcji wymiernej accurately.

The domain of a rational function is discussed, emphasizing that it excludes values that make the denominator zero. An example is provided: Df: R - {1; -2} for a specific function.

Vocabulary: The domain (dziedzina) of a rational function is the set of all possible input values that result in a real output.

Lastly, the page mentions that the graph of a rational function is typically a hyperbola, which is a key characteristic of these functions.

Highlight: The wykres funkcji wymiernej rysowanie process often involves identifying asymptotes and key points to sketch a hyperbola.