Precalculus Notes

Calculus is the mathematical study of change, and real-life things that change are modeled by functions. Precalculus is essentially the study of functions, with a few other related topics that supplement the study of functions and prepare students for calculus. In this text we'll concern ourselves first with learning about general functions, and later with certain types of common functions with special properties, like polynomial, exponential, logarithmic, and trigonometric functions. First, it is of critical importance to understand exactly what a function is. In the following lessons, we'll discuss what makes a function a function, some general properties of functions, and a few basic categories of functions. In this text we'll assume a general knowledge of algebraic principles of solving equations, working with the real numbers, and working with sets. In the last of the upcoming sections, we'll learn how functions behave under operations like addition, subtraction, etc. artesian Product - The set of all possible ordered pairs (a, b) composed of elements taken from the two sets, A and B .
Composition - A function operation symbolized (f o g)(x) that is equivalent to f (g(x)) .
Defined - A function is defined at a given value of the independent variable if it assigns that input an output; defined means "takes on a value".
Dependent Variable - The output variable of a function; the variable whose value depends on the input, or independent variable.
Domain - The set of all inputs for which a function or relation is defined.
Even Function - A function f is even if f (x) = f (- x) .
Function - A relation which assigns exactly one element in its range for each element in its domain.
Horizontal Line Test - The test by which it is shown whether a function is a one-to-one function or not, and therefore whether its inverse is a function.
Independent Variable - The variable of a function which does not depend on the other variable -- it is the