1. Defining and Representing Functions Math 1304 Calculus I 1. Defining and Representing Functions

Sections Covered 1.1: Four Ways to Represent a Function 1.2: Mathematical Models and Essential Functions 1.3: New Functions from Old

Four Ways to Represent a Function Covers functions: Section 1.1 Four Ways to Represent a Function Covers functions: Definition Terminology Conceptualization Ways to represent

Definition of Function (of Sets) Definition: A function is a rule that assigns to each element in one set exactly one element in another set. f(t) t

Terminology Domain – set of values for which the rule is defined Range – set of values that the rule produces as output Argument: input to the rule Value of: output from the rule Variables independent variable: input to the rule dependent variable: output from the rule

Conceptualization: arrow diagram f(x) x f(a) a A B

Ways to represent functions Verbally – use a language Numerically – use a table Visually – use a diagram Algebraically – use a formula Implicit: as formula that gives a relation between argument and value Explicit: value is given directly by a formula in terms of the argument

Examples See book for plenty of examples

Real Functions Note: In this case we study real-valued functions of a real variable. In other courses we study functions between other types of sets. Calculus III, functions can go from subsets of n-dimensional space to subsets of m-dimensional space. In Modern Algebra, functions often go between arbitrary finite sets. Sometimes they go between sets of whole numbers.

Graph of a Function The graph of a real-valued function of a real variable is a curve in the real plane.

Vertical Line Test Vertical line test – a curve in the xy-plane is the graph of a function if and only if no vertical line intersects the curve more than once. Not a function Is a function

Concepts Symmetry - odd or even functions – even functions satisfy: f(-x) = f(x) and odd functions satisfy: f(-x) = -f(x) Order - increasing/decreasing functions preserve or reverse order. Increasing: x < y f(x)<f(y) Decreasing: x < y f(x)>f(y)