Functions and Their Graphs College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved. Chapter 3 College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved. Chapter 3 Overview 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. Chapter 3 Objectives 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. Section 3.1 Functions 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. Function 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. Vertical Line Test Evaluate f x 1 given that f x x 3 x . 2 College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved. Common Mistake F x State the domain of the given function. 3 x 2 25 Solution : 3 F(x) Write the orginial equation. x Determine any restrictio ns on the values of x . Solve the restrictio n equation. State the domain Write domain The domain is restricted 25 25 0 2 25 or x 25 5 x 5 , 5 5 , 5 5 , notation. to all real numbers 2 x restrictio ns. in interval x 2 except 5 and 5. College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved. Domain of a Function Graphs of Functions; Piecewise-Defined Functions; Increasing and Decreasing Functions; Average Rate of Change Skills Objectives Conceptual 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. Identify common functions. Develop and graph piecewisedefined 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. Section 3.2 Graph the piecewise-defined function, and state the intervals where the function is increasing, decreasing, or constant, along with the domain and range. x 1 x f x 2 x 1 x 1 x 1 Click mouse to continue 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. x 1 x f x 2 x Increasing Decreasing Constant : 1, : , 1 : 1, 1 Domain : , 1 1, Range : 1, 1 x 1 x 1 College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved. Your Turn! Skills Objectives Conceptual 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 yaxis. Sketch the graph of a function by stretching or compressing a common function. Sketch the graph of a function using a sequence of transformations. 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. Section 3.3 Graphing Techniques: Transformations College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved. Vertical and Horizontal Shifts 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. Reflection About the Axes Use shifts and reflection State the domain to graph the function f x and range of f x Click mouse to continue x 1 2 College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved. Your Turn! Use shifts and reflection State the domain to graph the and range of f x Domain : 1, Range : - , 2 function f x x 1 2 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 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. Section 3.4 Operations on Functions and Composition of Functions College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved. Composition of Functions Given the functions Evaluate f x x 2 7 and g x 5 x 2 f g 1 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: f g 1 Write the desired quantity. Find the value of the inner function, Subsitute g. g 1 4 into f . g 1 5 1 4 2 f g 1 f 4 Evalue f 4 . f 4 4 7 9 2 f g 1 9 College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved. Evaluating a Composite Function 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. Section 3.5 One-to-One Functions and Inverse Functions 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

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