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Topic: Even and odd functions

	Even and odd functions - Wikipedia, the free encyclopedia
		The quotient of an even function and an odd function is an odd function.
		The derivative of an even function is odd.
		The composition of an even function and an odd function is even.
	en.wikipedia.org /wiki/Even_and_odd_functions (733 words)

	SparkNotes: Functions: Types of Functions
		The inverse of a function is the relation in which the roles of the independent anddependent variable are reversed.
		A one-to-one function is a function for which each element of the range corresponds to exactly one element of the domain.
		A piecewise function is a function which is defined by different rules depending on the value of the independent variable.
	www.sparknotes.com /math/precalc/functions/section3.rhtml (0 words)

	General Types of Functions
		In addition to their geometric and arithmetic properties, these transcendental functions are all solutions to "differential equations" -- equations relating the values of a function to its rate of change, the rate of change of the rate of change, and so on.
		In the discussion of asymptotes of rational functions we pointed out indirectly that every rational function is asymptotic to a polynomial - the one obtained by dividing through by the larges power of the variable in the denominator and setting all resulting negative powers to zero.
		Transcendental functions, on the other hand, can grow much faster (as in exponential functions) or much slower (as in logarithmic functions) than any power function, can be periodic (as in trigonometric functions), and can in general exhibit behavior beyond the algebraic realm.
	oregonstate.edu /instruct/mth251/cq/FieldGuide/general/lesson.html (0 words)

	SparkNotes: Functions, Limits, and Continuity: Brief Review of Functions
		A function is a rule that assigns to each element x from a set known as the "domain" a single element y from a set known as the "range".
		When two functions are combined so that the output of one function becomes the input for the other, the resulting combined function is called a composite function.
		One type of function we'll be dealing with often in calculus is the piecewise-defined function.
	www.sparknotes.com /math/calcab/functionslimitscontinuity/section1.html (0 words)

	BioMath: Functions
		To determine whether a function is increasing or decreasing we must “read” the graph of the function from left to right.
		Not all functions are classified as odd or even; a function that is not even or odd is called neither.
		If a function is bounded and has domain consisting of the entire real line, the global maximum is the highest peak and the global minimum is the lowest valley.
	www.biology.arizona.edu /BioMath/tutorials/Functions/Properties.html (0 words)

	Real & virtual energy. Semi-Circle & Pi. Transformation. Context as circle (Site not responding. Last check: 2007-10-19)
		The even function has infinitely inclusive properties such as those of a photon and it is an agreeable extension to call even functions 'feminine.' If the operation of translation or rotation have unique symmetries, then these operations can acquire gender as well.
		Generically, all even functions are forces [and may she be with you].
		Instead, because the circle is acceptable as both the even and the odd function, the circle is the originator and the go-between among the two.
	www.hyperflight.com /circle_and_pi.htm (0 words)

	Even and odd functions (Site not responding. Last check: 2007-10-19)
		In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking negatives.
		The sum of 2 odd functions is odd, and any constant multiple of an odd function is odd.
		In fact, the vector space of all real-valued functions is the direct sum of the spaces of even and odd functions.
	even-and-odd-functions.kiwiki.homeip.net (0 words)

	Even and Odd Functions
		IF the function in question is a polynomial then yes: a function is "even" if and only if all powers of x are even.
		A function is called "even" if f(-x)= f(x) for all numbers x (changing the sign on x doesn't change the value of the function at all), odd if f(-x)= -f(x) (changing the sign on x only changes the sign on f(x)) for all numbers x.
		f(x)=cos(x) for example is an even function while g(x)= sin(x) is an odd function.
	www.physicsforums.com /showthread.php?t=96250 (0 words)

	Symmetry of graphs - Topics in precalculus (Site not responding. Last check: 2007-10-19)
		A function symmetrical with respect to the y-axis is called an even function.
		A function that is symmetrical with respect to the origin is called an odd function.
		f(x) is even -- it is symmetrical with respect to the y-axis -- because f(−x) = f(x).
	www.themathpage.com /aPreCalc/symmetry.htm (0 words)

	MTH More on Functions (Site not responding. Last check: 2007-10-19)
		A function which Portia could use in her program to correctly calculate a customer’s rebate could be: Note: What formula to follows depends upon how much the customer charges on the card.
		Plug a number into the greatest integer function and the function replies with the greatest integer that is less than or equal to the value that you plugged in.
		Determine if the function is symmetric to the x-axis, to the y-axis, to the origin, or none of these.
	users.stlcc.edu /amosher/MoreonFunctions.htm (0 words)

	Even and odd numbers - Wikipedia, the free encyclopedia
		The parity of a permutation (as defined in abstract algebra) is the parity (even or odd) of the number of transpositions into which the permutation can be decomposed.
		An odd function, such as an odd power of a variable, gives the negative of its result when given the negative of an argument.
		It is possible for a function to be neither odd nor even, and for the case f(x) = 0, to be both odd and even.
	en.wikipedia.org /wiki/Even_and_odd_numbers (0 words)

	Visualization of parabolic numbers and odd and even functions
		The series assumes, in detail, the typical course of odd functions: this visual fact, on the base of the results summarized in Table 2, suggests the conjecture of the following paragraph.
		Even structure: infinite family of finite parabolas - Fig.
		Even structure: infinite family of finite parabolas Fig.
	members.tripod.com /vismath7/ippoliti/index.html (0 words)

	Graphs of Functions and Algebra - Interactive Tutorials
		Quadratic functions and the properties of their graphs such as vertex and x and y intercepts are explored interactively using an applet.
		An interactive large screen applet is used to explore logarithmic functions and the properties of their graphs such domain, range, x and y intercepts and vertical asymptote.
		Rational functions and the properties of their graphs such as domain, vertical and horizontal asymptotes, x and y intercepts are explored using an applet.
	www.analyzemath.com /precalculus.html (0 words)

	1.3 Fourier series of even and odd functions
		If a particular function possesses certain symmetrical properties then some terms are absent from its Fourier series expansion and the expressions determining the remaining coefficients can be simplified.
		The graph of an even function is symmetrical about the vertical axis and the graph of an odd function is symmetrical about the origin.
		Now, if f(x) is an odd function then integrand equals zero, and if f(x) is an even function then integrand equals 2f(x) and we have proved this lemma.
	math.ut.ee /~toomas_l/harmonic_analysis/Fourier/node4.html (0 words)

	All Elementary Mathematics - Study Guide - Functions and graphs - Basic notions and properties of functions...
		Now we can formulate a definition of a function more exactly: such a rule (law) of a correspondence between a set X and a set Y, that for each element of a set X one and only one element of a set Y can be found, is called a function.
		A graph of an even function is symmetrical relatively y-axis (Fig.5), a graph of an odd function is symmetrical relatively the origin of coordinates (Fig.6).
		Geometrically, a zero of a function is x-coordinate of a point of intersection of the function graph and x-axis.
	www.bymath.com /studyguide/fun/sec/fun6.htm (0 words)

	EMAT 6680 Class Page for Brian Deitz (Site not responding. Last check: 2007-10-19)
		The odd numbered functions produce a segemnt that produces sharper bends at 1 on the x and y axis.
		As you can see the bigger functions produced much sharper bends and half of each curve looks exactly the same for both the odd and even numbered exponents.
		With this project I was able to discover the difference in even and odd functions and what can happen when the exponent of those functions is increased.
	jwilson.coe.uga.edu /EMAT6680Su06/Deitz/Assignment1/assignment1.html (0 words)

	Some notes for Quantitative Methods, lecture 1
		We look here at even and odd functions; in the next lecture, we shall mention periodic functions.
		Functions can be classified as even, odd or neither.
		As illustrated in the lecture, you can tell by looking at its graph whether a function is even or odd.
	www.ee.surrey.ac.uk /Personal/J.Deane/Teach/qm1/lecs/lec1.html (0 words)

	3. Even and Odd Functions
		With knowledge of even and odd functions, a zero coefficient may be predicted without performing the integration.
		The graph of an even function is always symmetrical about the y-axis (a mirror image).
		The graph of an odd function is always symmetrical about the origin.
	www.intmath.com /Fourier/3_evenod.php (0 words)

	The function f (x) = xn, Even and Odd Functions
		y(- x) = - y(x) is called an odd function.
		Even function are symmetrical about the y-axis and odd functions are `skew' symmetrical about the y-axis.
		For example sin(x) is an odd function and cos(x) is an even function (N.B. most functions are neither even nor odd).
	www.maths.abdn.ac.uk /~igc/tch/ma1002/appl/node33.html (0 words)

	Visual Calculus - Even and Odd Functions (Site not responding. Last check: 2007-10-19)
		to recognize from the graph of a function whether a function is even, odd or neither; and
		An interactive LiveMath notebook to determine when a function is odd.
		A Quiz on whether a given function is even or odd.
	archives.math.utk.edu /visual.calculus/0/functions.14/index.html (0 words)

	Even and Odd Functions
		Some of the Fourier theorems can be succinctly expressed in terms of even and odd symmetries.
		An even function is also symmetric, but the term symmetric applies also to functions symmetric about a point other than 0.
		Example: For all DFT sinusoid is any function of the form A sin(ω t+φ), where t is the independent variable, and A, ω, φ are fixed parameters of the sinusoid called the amplitude, (radian) frequency, and phase, respectively.
	ccrma.stanford.edu /~jos/mdft/Even_Odd_Functions.html (0 words)

	fourier_even_odd_10_4.nb
		Because it is an even function, we consider its
		Because it is an odd function, we consider its
		Given a function defined on [0,L] for example, there are two ways to extend it over its complete period of 2L: as either an even or odd function.
	www.ireap.umd.edu /~nmoody/Math/fourier_even_odd.html (0 words)

	Function Notation: Even and Odd Functions
		You may be asked to determine algebraically whether a function is even or odd.
		This symmetry is a hallmark of odd functions.
		This function is the sum of the previous two functions.
	www.purplemath.com /modules/fcnnot3.htm (0 words)

	Functions and Transformations of Functions - HMC Calculus Tutorial
		The set of all possible values of f(x) in B is called the range of f.
		Here, we will only consider real-valued functions of a real variable, so A and B will both be subsets of the real numbers.
		The graph of an even function is symmetric about the y-axis, while the graph of an odd function is symmetric about the origin.
	www.math.hmc.edu /calculus/tutorials/transformations (0 words)

	Calculus: even and odd functions
		Is there any way that you can explain this even and odd funtion deal to me? My professor did not explain it very well and I am having a hard time understanding it.
		So any polynomial with only even exponents can be shown to be even.
		If you have even AND odd exponents mixed together, chances are your polynomial is NEITHER even nor odd.
	en.allexperts.com /q/Calculus-2063/odd-functions.htm (0 words)

	Graphing Techniques
		There are a number of concepts we will discuss in this section in order to learn how to sketch graphs: symmetry (even and odd functions), horizontal and vertical shifts, and vertical scaling (stretching and shrinking).
		These four functions are used in the examples and exercises to follow.
		When the actions described above are combined in one function we produce the resulting graph by determining the cumulative effect of all actions.
	colalg.math.csusb.edu /CAMDEMO/functions/src/manipulation.html (0 words)

	Powers of x
		Many of these functions will probably be quite familiar, but there are several subtle points that are worth remembering early on in a "Calculus career".
		The symmetry of the graphs of the power functions lends an important terminology to functions in general.
		Not every function is either even or odd.
	www.ugrad.math.ubc.ca /coursedoc/math100/notes/zoo/powers.html (0 words)

	Even and odd functions/Half-range expansions (Site not responding. Last check: 2007-10-19)
		Recall that a function is even if f(-x) = f(x) and odd if g(-x) = -g(x).
		For an even function we readily deduce that
		Meanwhile, for an odd function the symmetrical integral vanishes:
	www.soton.ac.uk /~jhr/MA273/node49.html (0 words)

	Even and Odd Functions
		Check all 8 functions f, g, h, i, j, k, l and m in the panel of the applet one by one and examine their graphs.
		The graphs that are symmetric with respect to y axis correspond to even functions and the graphs that are symmetric with respect to origin correspond to odd functions.
		+ 2 and is equal to g(x) hence function g is even.
	www.analyzemath.com /function/even_odd.html (0 words)

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