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Chapter 1. Functions and Models
Math 1304 Calculus I Chapter 1. Functions and Models
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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
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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
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Definition of Function
Definition: A function is a rule that assigns to each element in one set exactly one element in another set.
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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
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Conceptualization: arrow diagram
f(x) x f(a) a A B
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Conceptualization: Machine
Input Output
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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
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Examples See book for plenty of examples
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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.
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Graph of a Function The graph of a real-valued function of a real variable is a curve in the real plane.
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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
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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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