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Chapter 1: Functions and Graphs

Chapter 1: Functions and Graphs. Section 1-3: Twelve Basic Functions. Objectives. You will learn about: What graphs can tell us Twelve basic functions Analyzing functions graphically Why:

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Chapter 1: Functions and Graphs

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  1. Chapter 1:Functions and Graphs Section 1-3: Twelve Basic Functions

  2. Objectives • You will learn about: • What graphs can tell us • Twelve basic functions • Analyzing functions graphically • Why: • As you continue to study mathematics, you will find that the twelve basic functions presented here will come up again and again. By knowing their basic properties, you will recognize them when you see them.

  3. Vocabulary • Identity function • Squaring function • Cubing function • Reciprocal function • Square root function • Exponential function • Natural logarithm function • Sine function • Cosine function • Absolute value function • Greatest integer function • Logistic function

  4. Example 1Looking for Domains • Nine of the functions have domain the set of all real numbers. Which three do not? • One of the functions has domain the set of all real numbers except 0. Which function is it and why isn’t zero in its domain? • Which two functions have no negative numbers in their domains? Of these two, which one is defined at zero.

  5. Example 2Looking for Continuity • Only two of twelve functions have points of discontinuity. Are these points in the domain of the function?

  6. Example 3Looking for Boundness • Only three of the twelve functions are bounded (above and below). Which three?

  7. Example 4Looking for Symmetry • Three of the twelve basic functions are even. Which are they?

  8. Example 5Analyzing a Function • Graph the function y = (x - 2)2 • On what interval is the function increasing? On what interval is the function decreasing? • Is the function even, odd, or neither? • Does the function have any extrema? • How does the graph relate to the graph of the basic function y = x2

  9. Example 6Identifying a Piecewise Function • Which of the twelve basic functions has the following piecewise definition over separate intervals of its domain?

  10. Example 7Defining a Function Piecewise • Using basic functions from this section, construct a piecewise definition for the function whose graph is shown:

  11. Example 8Looking for a Horizontal Asymptote • Does the graph of y = ln x have a horizontal asymptote?

  12. Example 9Analyzing a Function • Give a complete analysis of the basic function f(x) = |x|. • Find: • Domain • Range • Continuous • Increasing and Decreasing • Symmetry • Boundness • Asymptotes • End Behavior

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