Rational Functions

Welcome to the course material on Rational Functions in Further Mathematics. Rational functions play a significant role in the realm of mathematics, particularly in the study of functions and their properties. This topic delves into the concept of rational functions, which are expressed as the ratio of two polynomials.

Understanding Rational Functions: At the core of rational functions is the expression of the form f(x) = g(x)/h(x), where g(x) and h(x) are polynomials. It is essential to grasp the idea that the functions involved are ratios of two polynomials. The degree of the numerator and denominator in a rational function holds paramount importance in analyzing its behavior.

Performing Operations on Rational Functions: In this course, you will learn to carry out fundamental operations such as addition, subtraction, multiplication, and division on rational functions. These operations involve the manipulation of the numerator and denominator of the rational functions according to established mathematical principles.

Resolution into Partial Fractions: A key aspect of rational functions is the process of resolving them into partial fractions. This technique is crucial in simplifying complex rational functions into more manageable components, aiding in further analysis and problem-solving.

Determining Domain and Range: Understanding the domain and range of rational functions is essential for comprehending the behavior of these functions. By identifying the restrictions on the input values (domain) and the corresponding output values (range), one gains insights into the overall function.

Identifying Zeros and Mapping Properties: The zeros of rational functions, which correspond to the values of x that make the function equal to zero, are significant points of interest. Moreover, exploring concepts like one-to-one and onto mappings, as well as determining the inverses of functions, enhances one's understanding of the structural properties of rational functions.

Graphical Analysis and Sketching: While graphical representations, such as sketching rational functions, are not mandatory in this course material, understanding the conceptual underpinnings of rational functions aids in visualizing their behavior and properties.

Logic and Syntactical Rules: Additionally, topics related to logic, syntax, and set theory will be covered to provide a comprehensive foundation for analyzing rational functions within a broader mathematical framework.

Through this course material, you will delve deep into the intricacies of rational functions, exploring their characteristics, properties, and applications in various mathematical contexts.