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Chapter 3

Chapter 3. Functions and Their Graphs. Chapter 3 Overview. Chapter 3 Objectives. Find the domain and range of a function. Sketch the graphs of common functions. Sketch graphs of general functions employing translations of common functions. Perform composition of functions.

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Chapter 3

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  1. Chapter 3 Functions and Their Graphs

  2. Chapter 3Overview

  3. Chapter 3Objectives • Find the domain and range of a function. • Sketch the graphs of common functions. • Sketch graphs of general functions employing translations of common functions. • Perform composition of functions. • Find the inverse of a function. • Model applications with functions using variation.

  4. Skills Objectives Determine whether a relation is a function. Determine whether an equation represents a function. Use function notation. Find the value of a function. Determine the domain and range of a function. Conceptual Objectives Think of function notation as a placeholder or mapping. Understand that all functions are relations but not all relations are functions. Section 3.1Functions

  5. Function A function is a correspondence between two sets where each element in the first set corresponds exactly to one element in the second set.

  6. Vertical Line Test Given a graph of an equation, if any vertical line that can be drawn intersects the graph at no more than one point, the equation defines y as a function of x. This test is called the vertical line test.

  7. Common Mistake

  8. Domain of a Function

  9. Skills Objectives Classify functions as even, odd, or neither. Determine whether functions are increasing, decreasing, or constant. Calculate the average rate of change of a function. Evaluate the difference quotient for a function. Graph piecewise-defined functions. Conceptual Objectives Identify common functions. Develop and graph piecewise-defined functions: Identify and graph points of discontinuity. State the domain and range. Understand that even functions have graphs that are symmetric about the y-axis. Understand that odd functions have graphs that are symmetric about the origin. Section 3.2 Graphs of Functions; Piecewise-Defined Functions; Increasing and Decreasing Functions; Average Rate of Change

  10. Your Turn! Graph the piecewise-defined function, and state the intervals where the function is increasing, decreasing, or constant, along with the domain and range. Click mouse to continue

  11. Your Turn! Graph the piecewise-defined function, and state the intervals where the function is increasing, decreasing, or constant, along with the domain and range.

  12. Skills Objectives Sketch the graph of a function using horizontal and vertical shifting of common functions. Sketch the graph of a function by reflecting a common function about the x-axis or y-axis. Sketch the graph of a function by stretching or compressing a common function. Sketch the graph of a function using a sequence of transformations. Conceptual Objectives Identify the common functions by their graphs. Apply multiple transformations of common functions to obtain graphs of functions. Understand that domain and range are also transformed. Section 3.3 Graphing Techniques: Transformations

  13. Vertical and Horizontal Shifts

  14. Reflection About the Axes The graph of –f(x)is obtained by reflecting the function f (x) about the x-axis. The graph of f(-x) is obtained by rotating the function f(x) about the y-axis.

  15. Your Turn! Click mouse to continue

  16. Your Turn!

  17. Vertical Stretching and Vertical Compressing of Graphs

  18. Horizontal Stretching and Horizontal Compressing of Graphs

  19. Skills Objectives Add, subtract, multiply, and divide functions. Evaluate composite functions. Determine domain of functions resulting from operations and composition of functions. Conceptual Objectives Understand domain restrictions when dividing functions. Realize that the domain of a composition of functions excludes the values that are not in the domain of the inside function. Section 3.4 OperationsonFunctionsandComposition of Functions

  20. Composition of Functions

  21. Evaluating a Composite Function Solution: One way of evaluating these composite functions is to calculate the two individual composites in terms of x: f(g(x)) and g(f(x)). Once those functions are known, the values can be substituted for x and evaluated. Another way of proceeding is as follows:

  22. Skills Objectives Determine whether a function is a one-to-one function. Verify that two functions are inverses of one another. Graph the inverse function given the graph of the function. Find the inverse of a function. Conceptual Objectives Visualize the relationships between the domain and range of a function and the domain and range of its inverse. Understand why functions and their inverses are symmetric about y =x. Section 3.5 One-to-One Functions and Inverse Functions

  23. Horizontal Line Test

  24. Inverse Functions

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