Chapter 2: Functions

A function is formally defined as a mapping that associates each input value from a domain with exactly one output value in the range, distinguishing it from general relations where inputs might correspond to multiple outputs. The chapter identifies two primary function types: one-one functions where each input produces a unique output matched to only one input, and many-one functions where multiple inputs can yield the same output. Operations on functions are explored through composition and inversion. Composite functions combine two functions sequentially, with function g acting on the input first and function f acting on that result, existing only when the range of g falls within the domain of f, and generally demonstrating non-commutativity. Inverse functions reverse the mapping of an original function, existing exclusively for one-one functions, with the domain and range swapping between f and f inverse. Graphically, inverse function pairs reflect across the line y equals x, a relationship that reveals self-inverse functions as symmetric about this line. The chapter then systematizes graph transformations into vertical and horizontal categories. Vertical transformations include translations determined by adding or subtracting constants, reflections across the x-axis through negation, and vertical stretching scaled by a multiplicative factor. Horizontal transformations involve translations parameterized by input modifications, reflections across the y-axis through input negation, and horizontal stretching with factors inversely related to the coefficient. A critical principle emerges regarding transformation order: vertical transformations follow standard operational hierarchy, while horizontal transformations follow the reverse, though combining one transformation of each type permits flexibility in sequencing. These concepts provide students with systematic methods for analyzing function behavior and predicting how algebraic modifications alter function graphs.