College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved. Chapter 3 Functions and Their Graphs

College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved. Chapter 3 Overview

College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved. 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.  Find the inverse of a function.  Model applications with functions using variation.

College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved. Section 3.1 Functions 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.

College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved. 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.

College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved. 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.

College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved. Common Mistake

College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved. Domain of a Function

College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved. Section 3.2 Graphs of Functions; Piecewise-Defined Functions; Increasing and Decreasing Functions; Average Rate of Change 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.

College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved. Your Turn! Click mouse to continue Graph the piecewise-defined function, and state the intervals where the function is increasing, decreasing, or constant, along with the domain and range.

College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved. 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.

College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved. Section 3.3 Graphing Techniques: Transformations 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.

College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved. Vertical and Horizontal Shifts

College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved. 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.

College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved. Your Turn! Click mouse to continue

College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved. Your Turn!

College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved. Vertical Stretching and Vertical Compressing of Graphs

College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved. Horizontal Stretching and Horizontal Compressing of Graphs

College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved. Section 3.4 Operations on Functions and Composition of Functions 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.

College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved. Composition of Functions

College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved. 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:

College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved. Section 3.5 One-to-One Functions and Inverse Functions 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.

College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved. Horizontal Line Test

College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved. Inverse Functions