CHAPTER 2: More on Functions 2.1 Increasing, Decreasing, and Piecewise Functions; Applications 2.2 The Algebra of Functions 2.3 The Composition of Functions 2.4 Symmetry and Transformations 2.5 Variation and Applications Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

2.1 Increasing, Decreasing, and Piecewise Functions; Applications Graph functions, looking for intervals on which the function is increasing, decreasing, or constant, and estimate relative maxima and minima. Given an application, find a function that models the application; find the domain of the function and function values, and then graph the function. Graph functions defined piecewise. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Increasing, Decreasing, and Constant Functions On a given interval, if the graph of a function rises from left to right, it is said to be increasing on that interval. If the graph drops from left to right, it is said to be decreasing. If the function values stay the same from left to right, the function is said to be constant. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Definitions A function f is said to be increasing on an open interval I, if for all a and b in that interval, a < b implies f(a) < f(b). Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Definitions continued A function f is said to be decreasing on an open interval I, if for all a and b in that interval, a < b implies f(a) > f(b). Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Definitions continued A function f is said to be constant on an open interval I, if for all a and b in that interval, f(a) = f(b). Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Relative Maximum and Minimum Values Suppose that f is a function for which f(c) exists for some c in the domain of f. Then: f(c) is a relative maximum if there exists an open interval I containing c such that f(c) > f(x), for all x in I where x  c; and f(c) is a relative minimum if there exists an open interval I containing c such that f(c) < f(x), for all x in I where x  c. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Relative Maximum and Minimum Values f y Relative minimum c1 c2 c3 x Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Applications of Functions Many real-world situations can be modeled by functions. Example A man plans to enclose a rectangular area using 80 yards of fencing. If the area is w yards wide, express the enclosed area as a function of w. Solution We want area as a function of w. Since the area is rectangular, we have A = lw. We know that the perimeter, 2 lengths and 2 widths, is 80 yds, P = 2l + 2w or 80 = 2l + 2w . If the width is w, then the length, l, can be given by 2l = 80 – 2w or l = 40 – w . Now A(w) = (40 – w)w = 40w – w2. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Functions Defined Piecewise Some functions are defined piecewise using different output formulas for different parts of the domain. For the function defined as: find f (-3), f (1), and f (5). Since –3 0, use f (x) = x2: f (–3) = (–3)2 = 9. Since 0 < 1 2, use f (x) = 4: f (1) = 4. Since 5 > 2 use f (x) = x – 1: f (5) = 5 – 1 = 4. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Functions Defined Piecewise Graph the function defined as: a) We graph f(x) = 3 only for inputs x less than or equal to 0. b) We graph f(x) = 3 + x2 only for inputs x greater than 0 and less than or equal to 2. f(x) = 3, for x  0 f(x) = 3 + x2, for 0 < x  2 c) We graph f(x) = only for inputs x greater than 2. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Functions Defined Piecewise Graph the function defined as: The graph of this part of the function consists of a line with a “hole” at (‒2, ‒4), indicated by the open circle. The hole occurs because a piece of the function is not defined for x = ‒2. f(‒2) = 3, so plot the point (‒2, 3) above the open circle. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Greatest Integer Function = the greatest integer less than or equal to x. The greatest integer function pairs the input with the greatest integer less than or equal to that input. -5 3 –6 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley