math 1304 calculus i chapter 1. functions and models
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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)
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