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Understanding Functions: Representation, Models, and Graphs

This chapter covers various aspects of functions including representation methods, mathematical models, composition, graphing, exponential and inverse functions. It explains the definition, terminology, and conceptualization of functions, along with ways to represent them verbally, numerically, visually, and algebraically. Real functions, symmetrical and increasing/decreasing functions are discussed, as well as the concept of the graph of a function and the vertical line test.

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Understanding Functions: Representation, Models, and Graphs

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  1. Math 1304 Calculus I Chapter 1. Functions and Models

  2. 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

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

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

  5. 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

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

  7. Conceptualization: Machine Input Output

  8. 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

  9. Examples • See book for plenty of examples

  10. 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.

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

  12. 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. Is a function Not a function

  13. 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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