Funkcja Wymierna and Advanced Function Properties
The second page delves into more advanced function concepts, primarily focusing on funkcja wymierna (rational functions) and further properties of various function types.
Funkcja wymierna is introduced as a function involving fractions of polynomials. The page emphasizes the importance of considering the domain of these functions, as they often have restrictions due to undefined values in the denominator.
Definition: A funkcja wymierna is a function of the form f(x) = P(x) / Q(x), where P(x) and Q(x) are polynomials and Q(x) ≠ 0.
The concept of asymptoty funkcji wymiernej (asymptotes of rational functions) is briefly mentioned, which is crucial for understanding the behavior of these functions at extreme values.
Highlight: When analyzing rational functions, it's essential to determine their domain and identify any asymptotes.
The page provides several examples of rational functions and demonstrates how to find their domains:
Example: For the function f(x) = 1 / (2-x), the domain is x ∈ ℝ - {2}, as x cannot equal 2 (which would make the denominator zero).
The topic of monotoniczność funkcji wymiernej (monotonicity of rational functions) is touched upon, emphasizing the importance of analyzing the function's behavior in different intervals.
The page also revisits the concept of miejsce zerowe funkcji for different function types, including quadratic and rational functions. It provides methods for finding these zeros, which are crucial for understanding a function's behavior and graph.
Vocabulary: Miejsce zerowe funkcji (function roots or zeros) are the x-values where the function equals zero, i.e., f(x) = 0.
Lastly, the page introduces the concept of graph transformations, particularly focusing on vertical and horizontal shifts of function graphs.
Example: For a function g(x) = f(x-p) + q, the graph of g is obtained by shifting the graph of f horizontally by p units and vertically by q units.
This section is particularly useful for understanding how changes in a function's equation affect its graphical representation, which is a key skill in mathematical analysis and problem-solving.
