what is the easy definition of a mathematical function?
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A function is a set of ordered pairs of numbers, no two of which have the same first element.
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Real Analysis is the branch of math that deals with functions of real variables.  This includes things like caluclus of real valued functions, metric spaces, and a little bit about the topology of R^n.
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In mathematics, a polynomial is an expression in which a finite number of constants and variables are combined using only addition, subtraction, multiplication, and non-negative whole number exponents (raising to a power).
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A derivative is a function that provides the limit of the slope of a section of a primary function as the length of that section approaches zero.
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Functions that maps x to x\nWhere x is an element of the function
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it can also be defined more geometrically this way:\n\nconsider the graph of\n\ny = 1/x\n\ne is the only point on the x-axis > 1 such that\nthe integral of y=1/x from x=1 to x=e is 1.\n\nin other words, for any number b >= 1, let A_b\nbe the area inside the region between x=1, x=b, y=0\n(the x-axis) and y=1/x. So for b=1, A_b = 0, and as b gets\nbigger, A_b gets bigger. In fact one can prove that\nA_b becomes as large as you want by making b big enough. So there's some point b=e where A_e = 1. That's e.\n\nThis is related to the logarithm, as the function\nf(b) = A_b\nis actually the natural log function, i.e.\nlog(b) = A_b.\n\nand, it's not a ratio of integers (not a rational number), i.e. there's no integers m, n such that e = m/n. this is also the case with pi. i guess when you say pi is a ratio, you mean the ratios to do with the circle. e is a ratio in that sense, for instance e = e/1, or e = (pi*e)/pi. but i'm not aware of any useful geometric ratio.\n\ne and pi also have the property (along with most real numbers) that there's no polynomial with integer coefficients p(x) such that p(e)=0 or p(pi)=0. contrast this with say sqrt(11), which isn't a rational number, but does give 0 when plugged in the polynomial\n\nP(x) = x^2 - 11.\n\nthis is the definition of transcendental versus algebraic. e and pi are trans, whereas sqrt(11) is algebraic. there are "more" transcendental numbers than algebraic numbers, in a quite reasonable sense of "more".
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Main Entry: in·for·ma·tion \nPronunciation: "in-f&r-'mA-sh&n\nFunction: noun\n1 : the communication or reception of knowledge or intelligence\n2 a (1) : knowledge obtained from investigation, study, or instruction (2) : INTELLIGENCE, NEWS (3) : FACTS, DATA b : the attribute inherent in and communicated by one of two or more alternative sequences or arrangements of something (as nucleotides in DNA or binary digits in a computer program) that produce specific effects c (1) : a signal or character (as in a communication system or computer) representing data (2) : something (as a message, experimental data, or a picture) which justifies change in a construct (as a plan or theory) that represents physical or mental experience or another construct d : a quantitative measure of the content of information; specifically : a numerical quantity that measures the uncertainty in the outcome of an experiment to be performed\n3 : the act of informing against a person\n4 : a formal accusation of a crime made by a prosecuting officer as distinguished from an indictment presented by a grand jury\n- in·for·ma·tion·al  /-shn&l, -sh&-n&l/ adjective\n- in·for·ma·tion·al·ly adverb
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Mathematics is not only functional but also beautiful.  Did you know that you can get a Math degree in Science or Arts?\n\nAnyway balancing an equation or putting it in its simplest form means removing all the extraneous and distracting information so that you can see the core idea more clearly.\n\nA simple example:  Lets look at the following equation.  13x=208  At first glance it is hard to see that x=16 so we want to simplify the equation.  The equals sign is kinda like a fulcrum or a pivot point.  If you do the same thing to both sides of the equation then it stays balanced (and stays true)  In this case divide both sides by 13 and you can see now that x=6.  \n\nFor the two equations 13x=208 and x=6.  They have the same meaning but one is easier to understand because it is in a simpler form.  To get from one to the other we have to maintain balance.
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A fascinating subject.\n\nIt's recursively defined.  \n\nf(n+2) = f(n+1) + f(n) for n>0\n\nwhere f(1) = f(2) = 1.\n\nIt has some really neat properties and shows up in nature.
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A linear function can refer to two slightly different concepts. In geometery and elementary algebra a linear function is a first degree polynomial mathematical function of the form:\n\nf(x) = m x + c \nwhere m and c are constants.\n\nThe problem with this geometric definition is that functions of the above form - despite their names - do not necessarily satisfy the conditions of a linear map. Therefore, some people refer to functions of the above form as affine functions. If and only if a function is of the above form with c equal to zero, the function satisfies the properties of a linear map, preserving scalar multiplication and vector addition for all points in its domain.\n\nLinear functions always have as domain the set of all real numbers and a range of all real numbers. By the geometric definition, the derivative of linear functions in terms of their independent variable, x, is always the constant m.\n\nLinear functions (according to the geometric definition) can also be written in the form:\n\ny = m x + c \nand plotted on an x,y graph. It forms a straight line, as the name implies.\n\nThe constant m is often called the slope or gradient while c is the y-intercept, which gives the point of intersection between the graph of the function and the y-axis.
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In mathematics the term "periodic function" is strictly defined as a function that repeats itself after a fixed increment of the independent variable.  The increment of the independent variable that defines the "repeat time" is called the period of the function.  \n\nFor example, sin(x) and cos(x) are periodic functions because the values of sin(x) and cos(x) repeat after every 360 degrees (or 2*pi radians if you prefer those units for x).\n\nIn fields other than mathematics, the term periodic function is sometimes used more loosely, as in the chemical example you were given as a clue.  The chemical and physical properties are not really a true periodic function of their atomic numbers (BTW, tell your teacher that it's the atomic *number* not the atomic *mass* that's important for determining the properties of an element) because: (1) the properties of the elements are not *exactly* the same, but merely similar; (2) the period after which the properties sort of repeat is not a constant -- it takes different numbers of electrons to fill the available suborbitals of the elements as one moves down the rows of the periodic table.\n\nOther examples of periodic functions of time (loosely defined) would include your blood pressure, the height above the horizon of the sun, the phase of the moon, etc.
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In calculus, the integral of a function is a generalization of the concept of a sum, typically with direct physical interpretation pertaining to area, mass, or volume. The process of finding integrals is integration, in its mathematical meaning. Unlike the closely-related process of differentiation, there are several possible definitions of integration, with different technical underpinnings. They are, however, compatible; any two different ways of integrating a function will give the same result when they are both defined.\n\nThe word "integral" may also refer to antiderivatives in a mild abuse of language. (The antiderivative of a function is that function whose derivative is equal to the first function.) Though they are closely related through the fundamental theorem of calculus, the two notions are conceptually distinct. When one wants to clarify this distinction, an antiderivative integral is referred to as an indefinite integral (a function), while the integrals discussed in this article are termed definite integrals.
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You're probably talking about the vertical line test.  A relation is a function iff you can draw a vertical line anywhere and intersect the function at just one point.\n\nSo, something like y = x is a function because a vertical line always hits just one point.  y^2 = x is not a function because if you draw a vertical line at x=4, it hits the graph at both (4,2) and (4,-2).
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"Maths" is a contraction of "mathematics", a short way to say and write it.  It includes all branches of maths, e.g. arithmetic, algebra, geometry, trigonometry, statics, differential and integral calculus, quantum mechanics, etc. As you can see, arithmetic is a subset of mathematics.  Arithmetic deals with the manipulation of real numbers only.
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Math is the language of Science
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Generally speaking, you call "linear" a function F (or an operator whatsoever) which satisfies the following conditions:\n\n1) F(a*X)=a*F(X) for every real number a (X is the argument of the function, whatever it be: it can be a number, a vector, a complex, a function... it doesn't matter)\n\n2) F(X+Y) = F(X)+F(Y) for every X and Y which can be taken as an argument of the function or the operator.\n\nThese two conditions can be sinthesized in the following single one:\n\nCONDITION FOR LINEARITY:\nF(a*X+b*Y) = a*F(X) + b*F(Y) for every a and b reals and for every X and Y in the domain of the function.\n\n----\n\nIn the ring of the polynomials of n variables x1, x2 ... xn, it is easily verified that the only polynomials which can be defined "linear" according to the above conditions are those in which the variables appear at the first degree only, so they are in the form:\n\nP(n) = a1*x1 + a2*x2 + ... + an*xn + a(n+1)\n\nLinear equations in n variables are obtained by setting a linear polynomial of the above type equal to 0.\n\nNon-linear equations are all other possible kinds of equations: the ones obteined by setting to 0 non-linear polynomials (you call them algebraic non-linear equations), or the ones obtained by setting ot 0 expressions which involve logarithms, exponentials etc (in that case you are dealing with transcendant equations)
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This would most definitely be a polynomial function, since it starts with x cubed. Anything that has x cubed or higher in it, like x to the fifth power, would be a polynomial function. An equation with x squared would be a quadratic function. An equation with just x would be a linear equation.