Even Functions

7 Key excerpts on "Even Functions"

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  eBook - PDF

  Algebra and Trigonometry

  • Sheldon Axler(Author)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  152 chapter 3 Functions and Their Graphs Even Functions Suppose f (x) = x 2 for every real number x. Notice that f (-x) = (-x) 2 = x 2 = f (x). This property is sufficiently important that we give it a name. Even Functions A function f is called even if f (-x) = f (x) for every x in the domain of f . For the equation f (-x) = f (x) to hold for every x in the domain of f , the expression f (-x) must make sense. Thus -x must be in the domain of f for every x in the domain of f . For example, a function whose domain is the interval [-3, 5] cannot possibly be an even function, but a function whose domain is the interval (-4, 4) may or may not be an even function. As we have already observed, x 2 is an even function. Here is another simple example: example 10 Show that the function f defined by f (x) = |x| for every real number x is an even 3 3 x 3 y The graph of |x| on the interval [-3, 3]. function. solution This function is even because f (-x) = | - x| = |x| = f (x) for every real number x. Suppose f is an even function. As we know, flipping the graph of f across the vertical axis gives the graph of the function h defined by h(x) = f (-x). Because f is even, we actually have h(x) = f (-x) = f (x), which implies that h = f . In other words, flipping the graph of f across the vertical axis gives us back the graph of f . Thus the graph of an even function is symmetric about the vertical axis. This symmetry can be seen, for example, in the graph shown above of |x| on the interval [-3, 3]. Here is the statement of the result in general: The graph of an even function A function is even if and only if its graph is unchanged when flipped across the vertical axis. section 3.2 Function Transformations and Graphs 153 Odd Functions Consider now the function defined by f (x) = x 3 for every real number x. Notice that f (-x) = (-x) 3 = -(x 3 ) = -f (x). This property is sufficiently important that we also give it a name.

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  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.

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  Foundations of Mathematics

  Algebra, Geometry, Trigonometry and Calculus

  • Philip Brown(Author)
  • 2016(Publication Date)
  • Mercury Learning and Information

    (Publisher)

  EXAMPLE 5.9.1. In figure 5.16, the graphs on the left are the graphs of Even Functions, and the graphs on the right are the graphs of odd functions. ± Functions • 141 y = x 2 y = x 3 y = cos(x) y = sin(x) y = | x | y = x FIGURE 5.16. Even and odd functions. REMARK 5.9.2. A function is an even function if and only if it has the algebraic property f x f x ( ) ( ) − = for any value x in the domain of f, and a function is an odd function if and only if, it has the algebraic property f x f x ( ) ( ) − = − for any value x in the domain of f. EXAMPLE 5.9.2. = + f x x x ( ) 5 is an odd function because f x x x x x x x f x ( ) ( ) ( ) ( ) ( ) − = − + − = − − = − + = − 5 5 5 ± 5.10 OPERATIONS ON FUNCTIONS Functions can be regarded as elements of an algebraic system, meaning that we can add, subtract, multiply, and divide them. We can also operate on functions by forming compositions of functions. 5.10.1 The Algebra of Functions DEFINITION 5.10.1. Suppose that f and g are functions with domains A and B, respectively, then we can create the new functions. f g f g fg f g + − , , , / , defined by  + = + ∈ ∩ - = - ∈ ∩ = ⋅ ∈ ∩       = ∈ ∩ ∈ = f g x f x g x x A B f g x f x g x x A B fg x f x g x x A B f g x f x g x x A B x g x ( )( ) ( ) ( ), for ( )( ) ( ) ( ), for ( )( ) ( ) ( ), for ( ) ( ) ( ) , for /{ | ( ) 0}. The next example demonstrates that it is possible to prove statements about the properties of functions, in general (i.e., statements that do not require specification of any particular functions). 142 • Foundations of Mathematics EXAMPLE 5.10.1. Prove that a sum of odd functions is an odd function.

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  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.

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  The Calculus Lifesaver

  All the Tools You Need to Excel at Calculus

  • Adrian Banner(Author)
  • 2009(Publication Date)
  • Princeton University Press

    (Publisher)

  This simplifies to 2 f (0) = 0, or f (0) = 0 as claimed. Anyway, starting with a function f , how can you tell if it is odd, even, or neither? And so what if it is odd or even anyway? Let’s look at this second question before coming back to the first one. One nice thing about knowing that a function is odd or even is that it’s easier to graph the function. In fact, if you can graph the right-hand half of the function, the left-hand half is a piece of cake! Let’s say that f is an even function. Then since f ( x ) = f ( -x ), the graph of y = f ( x ) is at the same height above the x -coordinates x and -x . This is true for all x , so the situation looks something like this: Section 1.4: Odd and Even Functions • 15 x Same height -x We can conclude that the graph of an even function has mirror sym-metry about the y -axis . So, if you graph the right half of a function which you know is even, you can get the left half by reflecting the right half about the y -axis. Check the graph of y = x 2 to make sure that it has this mirror symmetry. On the other hand, let’s say that f is an odd function. Since we have f ( -x ) = -f ( x ), the graph of y = f ( x ) is at the same height above the x -coordinate x as it is below the x -coordinate -x . (Of course, if f ( x ) is negative, then you have to switch the words “above” and “below.”) In any case, the picture looks like this: 16 • Functions, Graphs, and Lines In the example above, you’d write f ( -x ) = log 5 (2( -x ) 6 -6( -x ) 2 + 3) = log 5 (2 x 6 -6 x 2 + 3) , which is actually equal to the original f ( x ). So the function f is even. How about g ( x ) = 2 x 3 + x 3 x 2 + 5 and h ( x ) = 2 x 3 + x -1 3 x 2 + 5 ? Well, for g , we have g ( -x ) = 2( -x ) 3 + ( -x ) 3( -x ) 2 + 5 = -2 x 3 -x 3 x 2 + 5 . Now you have to observe that you can take the minus sign out front and write g ( -x ) = -2 x 3 + x 3 x 2 + 5 , which, you notice, is equal to -g ( x ). That is, apart from the minus sign, we get the original function back.

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  CounterExamples

  From Elementary Calculus to the Beginnings of Analysis

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

    (Publisher)

  A function is bounded if it is bounded above and below. Otherwise, a function is unbounded . Even function . A function f ( x ) defined on X is called even if for any x ∈ X it holds that f ( − x ) = f ( x ). Odd function . A function f ( x ) defined on X is called odd if for any x ∈ X it holds that f ( − x ) = − f ( x ). Periodic function . A function f ( x ) defined on X is called periodic with period T ̸ = 0 if for any x ∈ X it holds that f ( x + T ) = f ( x ). The smallest positive number T (if it exists) for which this property holds is called the fundamental period. The primarily (geometric) properties of Even Functions (follow directly from the definition): 1) the domain of an even function is symmetric with respect to the origin; 2) the graph of an even function is symmetric with respect to the x -axis. The primarily properties of odd functions (follow directly from the defini-tion): 1) the domain of an odd function is symmetric with respect to the origin; 2) the graph of an odd function is symmetric with respect to the origin. Elementary properties of functions 5 The primarily properties of periodic functions (follow directly from the definition): 1) the domain of a periodic function is not bounded either at the left or at the right; 2) the graph of a periodic function can be obtained by an infinite many trans-lations of its part defined on an interval of length T . Increasing function . A function f ( x ) is called increasing (strictly in-creasing) on a set S if for any x 1 , x 2 ∈ S , x 1 < x 2 it follows that f ( x 1 ) ≤ f ( x 2 ) ( f ( x 1 ) < f ( x 2 ) ). Decreasing function . A function f ( x ) is called decreasing (strictly de-creasing) on a set S if for any x 1 , x 2 ∈ S , x 1 < x 2 it follows that f ( x 1 ) ≥ f ( x 2 ) ( f ( x 1 ) > f ( x 2 ) ). Monotone function . A function f ( x ) is called monotone (strictly mono-tone) on a set S if it is increasing or decreasing (strictly increasing or strictly decreasing) on this set.

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  Guide to Mathematical Methods

  • John Gilbert, Camilla Jordan, David A Towers(Authors)
  • 2017(Publication Date)
  • Red Globe Press

    (Publisher)

  3. For Examples 2.2(3), we can only define the function T : A → R , where the domain A of T is the set of possible dates, by giving a table of values T ( t ) for every t in A . As we have seen, many functions can be defined by a formula; such a formula can often be expressed in terms of a few basic functions, which we shall now describe. Later we shall combine these basic functions in various ways to gen-erate a whole collection of other functions, which we shall call the family of standard functions . Before we leave this section we introduce some useful terminology that is often used with functions. 24 Guide to Mathematical Methods x y x y x y Figure 2.3: Graphs of y = x 2 , y = x, y = sin x Definition 2.4 Let f : A → R where A ⊆ R . Then • f is said to be an even function if f ( x ) = f ( − x ) for all x ∈ A ; • f is said to be an odd function if f ( x ) = − f ( − x ) for all x ∈ A ; • f is said to be a periodic function if there is an a ∈ R such that f ( x + a ) = f ( x ) for all x ∈ R . The period of f is a . Notice that the graph of an even function will be symmetric about the y -axis and the graph of an odd function will have rotational symmetry of order 2. Examples 2.4 See Figure 2.3. 1. f ( x ) = x 2 is an even function. 2. f ( x ) = x is an odd function. 3. f ( x ) = sin x is a periodic function since sin( x + 2 π ) = sin x for all x ∈ R . [If you have not met the sin function before it is introduced in a later section in this chapter.] Functions 25 Exercises: Section 2.2 1. Which of the following rules define y as a function of x ? Give reasons for your answers. (You may need to refer to a later section in this chapter if you have not already met tan or cos.) (i) For each number x, y is given by tan y = x . (ii) For each number x, y satisfies y 2 = x 2 + 1. (iii) For each number x, y = | x | (this is the modulus function defined by y = x if x ≥ 0 , y = − x if x < 0).

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