Math 1304 Calculus I

Chapter 1. Functions and Models

Sections Covered in Chapter 1

• 1.1: Four Ways to Represent a Function• 1.2: Mathematical Models and Essential

Functions • 1.3: New Functions from Old - composition• 1.4: Graphing functions• 1.5: Exponential Functions• 1.6: Inverse Functions

Section 1.1

• Four Ways to Represent a Function

• Covers functions:– Definition

– Terminology

– Conceptualization

– Ways to represent

Definition of Function

• Definition: A function is a rule that assigns to each element in one set exactly one element in another set.

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

A B

f(a)a

Conceptualization: Machine•

Input Output

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)