Odd functions

5 Key excerpts on "Odd functions"

• 

  eBook - PDF

  Precalculus

  • Cynthia Y. Young(Author)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  The term even is used to describe functions that are symmetric with respect to the y-axis, or vertical axis, and the term odd is used to describe functions that are symmetric with respect to the origin. Recall from Section 0.5 that symmetry can be determined both graphically and algebraically. The box below summarizes the graphical and algebraic characteristics of even and Odd functions. x y (–8, –2) (8, 2) 10 5 Cube Root Function Domain: (–`, `) Range: (–`, `) x y (2, 2) (–2, 2) Absolute Value Function Domain: (–`, `) Range: [0, `) x (1, 1) (–1, –1) y Reciprocal Function Domain: (–, 0)  (0, ) Range: (–, 0)  (0, ) CUBE ROOT FUNCTION ƒ 1 x 2 5 ! 3 x or ƒ 1 x 2 5 x 1/3 ABSOLUTE VALUE FUNCTION ƒ 1 x 2 5 | x | RECIPROCAL FUNCTION ƒ 1 x 2 5 1 x x 2 0 EVEN AND Odd functions Function Symmetric with Respect to On Replacing x with 2x Even y-axis, or vertical axis ƒ 1 2x 2 5 ƒ 1 x 2 Odd origin ƒ 1 2x 2 5 2ƒ 1 x 2 The algebraic method for determining symmetry with respect to the y-axis, or vertical axis, is to substitute in 2x for x. If the result is an equivalent equation, the function is symmetric with respect to the y-axis. Some examples of even functions are ƒ 1 x 2 5 b, ƒ 1 x 2 5 x 2 , ƒ 1 x 2 5 x 4 , and ƒ 1 x 2 5 | x |. In any of these equations, if 2x is [ CONCEPT CHECK] Classify the functions f 1 x2 5 x 2n and g 1 x2 5 x 2n11 , where n is a positive integer (1, 2, 3, …), as even, odd, or neither. ANSWER f 1 x2 is even; g 1 x2 is odd ▼ 1.2 Graphs of Functions 121 122 CHAPTER 1 Functions and Their Graphs substituted for x, the result is the same, that is, ƒ 1 2x 2 5 ƒ 1 x 2 . Also note that, with the exception of the absolute value function, these examples are all even-degree polynomial equations. All constant functions are degree zero and are even functions. The algebraic method for determining symmetry with respect to the origin is to substitute 2x for x.

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Symmetric Properties of Real Functions

  • Brian thomson(Author)
  • 2020(Publication Date)
  • CRC Press

    (Publisher)

  6 Odd Properties

  6.2 Introduction

  In this chapter we complete a systematic study of the odd properties of real functions. We recall that the study of the odd properties of a function reveals its even symmetric structure. At the simplest level a function is even if and only if its odd part vanishes. Similarly a greater or lesser degree of control on the behavior of the odd part of a function expresses some characteristic of its even symmetric structure.

  The symmetry properties that we propose now to study all concern the nature of a function f that satisfies some requirement on the size of the expression

  f

  ( x + h )

  − f

  ( x − h )

  ( as h → 0 )

  .

  For example the most severe requirement is to ask that at each point x there is a neighbourhood in which the symmetry f (x + h ) = f (x − h ) holds. We shall say a function f is exactly locally symmetric at x if this holds. One expects that this vanishing odd part, giving a local even symmetry at each point, should arise only for a function that is symmetric about every vertical line (i.e. a constant function). This is very nearly the case and leads to some interesting results. A much weaker requirement might be to ask only that

  ( f

  ( x + h )

  − f

  ( x − h )

  ) / 2 h

  remain bounded as h → 0; a function with a symmetric derivative would have this property. We have already encountered some results of this type in the first two chapters and we have seen a large collection of analogous problems stated in terms of even properties in Chapter 4 . While there are parallels in the behavior of the odd and even properties our study reveals a broad range of differences and introduces different techniques.

  Properties relating more directly to the symmetric derivatives themselves will be studied in Chapter 7 .

  6.2 Symmetry

  We begin our study with exact symmetry conditions. A set E would be exactly symmetric at a point x if the condition x + h ∈ E is equivalent to x — h ∈ E . A function would be exactly symmetric at a point x if there is an identity f (x +h ) = f (x −h

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Explorations in College Algebra

  • Linda Almgren Kime, Judith Clark, Beverly K. Michael(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  A function f (x) is odd if f (−x) = − f (x). a. Using what you know about reflections and symmetry, describe how the graphs of even and Odd functions differ. b. Are the following functions even, odd, or neither? i. N(x) = x 3 iii. M(x) = 5 x ii. F(x) = x 2 iv. G(x) = | x | 19. Use the definition of even and Odd functions in Exercise 18 to verify the following statements. a. The sum of two even functions is an even function. b. The sum of two Odd functions is an odd function. c. The product of two even functions is an even function. d. The product of two Odd functions is an even function. e. The product of an even function and an odd function is an odd function. 20. The accompanying graph gives the annual sales S(t) and profit (or loss) P(t) for Apple for year t from 2002 to 2009. a. What would S(t) − P(t) represent? Describe the part of the graph that represents S(t) − P(t). b. What would [P(t)/S(t)] ⋅ 100% represent? 40 35 30 25 20 15 –5 10 5 0 Billions $ P(t) S(t) t 2001 2002 2003 2004 2005 2006 2007 2008 2009 Sales billions $ Profit (loss) billions $ Apple Annual Financial Data All Segments Source: investor.apple.com 21. In Section 1.2, Exercise 1, there is a graph about AIDS diagnoses and deaths from 1981 to 2007. If t = year, A(t) = number of people diagnosed with AIDS, and D(t) = number of people who died from AIDS, both in year t, what would A(t) − D(t) represent? How could you depict this on the graph? 22. When considering a career path in a particular job sector, one might examine the growth (or decline) of that sector of the job mar- ket. The following graphs illustrate the growth in Education & Health Services and the Leisure & Hospitality sector, compiled by the Fed- eral Reserve.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  CounterExamples

  From Elementary Calculus to the Beginnings of Analysis

  • Andrei Bourchtein, Ludmila Bourchtein(Authors)
  • 2014(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Example 3 . “If the sum of two functions is odd, then each of these two functions is also odd.” Solution. Let f ( x ) = { x, x < 1 − x, x ≥ 1 and g ( x ) = { x, x < 1 3 x, x ≥ 1 . Both functions are neither even nor odd, but h ( x ) = f ( x ) + g ( x ) = 2 x is an odd function. Remark 1 . A similar simple example can be given for the difference. Elementary properties of functions 21 Remark 2 . The converse is correct: the sum and difference of two Odd functions is again an odd function (when it is defined). Remark 3 . For the product and ratio, if each of two functions is odd then the result (when defined) is an even function. Therefore, it is hard even to suppose that the product (or the ratio) of two Odd functions will give another odd function. FIGURE 1.5.2 : Example 3 Example 4 . “If the product of two functions is odd, then one of these functions is even and another is odd.” Solution. Let f ( x ) = { x, x < 1 − x, x ≥ 1 and g ( x ) = { 1 , x < 1 − 1 , x ≥ 1 . Both functions are neither even nor odd, but h ( x ) = f ( x ) · g ( x ) = x is an odd function. Remark 1 . A similar simple example can be given for the ratio. Remark 2 . The converse is correct: the product and ratio of an even function and an odd function is an odd function (when it is defined). Example 5 . “If the absolute value of a function is even, then the function itself is also even.” Solution. If f ( x ) = x (odd function), then | f ( x ) | = | x | is even function. 22 Counterexamples: From Calculus to the Beginnings of Analysis Remark 1 . Of course, the original function can be neither even nor odd: f ( x ) = { x, x < 1 − x, x ≥ 1 gives the same even function | f ( x ) | = | x | . Remark 2 . The same counterexamples can be used to the following false statement: “if the square of a function is even, then the function itself is also even”. Remark 3 . The converse is true: if a function is even, then its absolute value and its square are also even functions.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  A Course of Mathematical Analysis

  International Series of Monographs on Pure and Applied Mathematics

  • A. F. Bermant, I. N. Sneddon, S. Ulam, M. Stark(Authors)
  • 2016(Publication Date)
  • Pergamon

    (Publisher)

  In an interval of positive sign of a function, its graph is located above Ox, whilst it is below Ox in an interval of negative sign; at a zero of the function, the graph has a common point with Ox (Fig. 5). II. Definition. A function y =y*(#), defined in an interval ( — a, a), is described as even if, on changing the sign of any value of the argu- FUNCTIONS 41 ment belonging to this interval, the value of the function is unchanged: A function y = / ( * ) , defined in an interval ( — a, a), is described as odd if, on changing the sign of any value of the argument belonging to this interval, only the sign (and not the absolute value) of the function is changed: / < -* ) = -/<*)· Examples of even functions are y = x 2 , y = cos x, whilst y = x 3 , y = sin x are examples of Odd functions. -x o y* /ΝΠ Even kr o x x Odd FIG. 6 The graph of an even function is symmetrical with respect to Oy; the graph of an odd function is symmetrical with respect to the origin. (We leave the reader to prove this; see Fig. 6.) Obviously, a function may be neither even nor odd; for example, y = x + 1, y = 2 sin x -j- 3 cos x 9 y = 2 X and so on are such func-tions. III. Definition. A function y =f(x) is said to be periodic if there exists a constant number a φ 0 such that, on adding it to any value of the argument, the value of the function is unchanged: /(* + «) =/(*>. If a function is periodic, the following equalities also hold: f(x + 2a) = fix), f(x + 3a) = f(z), f(x-a) = / ( * ) , f(x-2a) = f(x) and in general: f{x + ka) =f(x) for any x and for any integer k (positive or negative). 42 COUBSE OF MATHEMATICAL ANALYSIS Definition. The period of a function is the least positive number which, when added to any value of the argument, leaves the function unchanged. An example of a periodic function is y = sin x; its period is equal to 2π.

  Sign up to read

  Learn more about book