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Functional and Graph

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1.0 INTRODUCTION

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable. The concept of function was formalized at the end of the 19th century in terms of set theory, and this greatly enlarged the domains of application of the concept.

A function is uniquely represented by its graph which is the set of all pairs (xf (x)) for getting a global view of its properties. Some functions may also represented histograms. When the domain and the codomain are sets of numbers, each such pair may be considered as the Cartesian coordinates of a point in the plane. In general, these points form a curve, which is also called the graph of the function. This is a useful representation of the function, which is commonly used everywhere, for example in newspapers.

A specific function is, in general, defined by associating to every element of its domain one element of its codomain. When the domain and the codomain are sets of numbers, this association may take the form of a computation taking as input any element of the domain and producing an output in the codomain. This computation may be described by a formula. This is the starting point of algebra, where many similar numerical computations can be replaced by a single formula that describes these computations by means of variables that represent computation inputs as unspecified numbers. This type of specification of a function frequently uses previously defined auxiliary functions.

A computation that defines a function may often be described by an algorithm, and any kind of algorithm may be used. Sometimes, the definition of a function may involve elements or properties that can be defined, but not computed.

Functions are widely used in science, and in most fields of mathematics. Their role is so important that it has been said that they are "the central objects of investigation" in most fields of mathematics.

2.0 FUNCTIONS

A function is a process or a relation that associates each element x of a set X, the domain of the function, to a single element y of another set Y (possibly the same set), the codomain of the function. If the function is called f, this relation is denoted y = f (x) (read f of x), the element x is the independent variable or input of the function, and y is the value of the function, the output, or the dependent variable of x by f. The symbol that is used for representing the input is the variable of the function (one often says that f  is a function of the variable x).

In the definition of function, X  and Y  are respectively called the domain and the codomain of the function f. If (x, y) belongs to the set defining f, then y is the image of x under f, or the value of f  applied to the argument x. Especially in the context of numbers, one says also that y is the value of f  for the value x of its variable, or, still shorter, y is the value of f  of x, denoted as y = f(x).

Thus, the output f(x) is the same as y. But since y = x+2, we can also write f(x) = y = x+2 or more simply f(x) = x + 2. For example, to find f (4), which is the output corresponding to the input 4, we replace each x  in f (x) = x+2 by 4: f (4) = 4 + 2 = 6. Outputs are also called function values. Intuitively, a function is a process that associates to each element of a set X a unique element of a set Y.

One of the more important ideas about functions is that of the domain and range of a function. In simplest terms the domain of a function is the set of all values that can be plugged into a function and have the function exist and have a real number for a value. So, for the domain we need to avoid division by zero, square roots of negative numbers, logarithms of zero and logarithms of negative numbers. The range of a function is simply the set of all possible values that a function can take.

2.1 SPECIAL FUNCTIONS

Constant Function

A constant function is a function that has the same output value no matter what your input value is. Because of this, a constant function has the form y = b, where b is a constant (a single value that does not change). For example, y = 7 or y = 1,094 are constant functions. No matter what input, or x-value is, the output, or y-value is always the same.

Polynomial Function

The polynomial is named or identitied by the term of the highest power.

constant term: a0x0 or a0

linear term: a1x1

quadratic term: a2x2

Rational Function

A rational function is a function that is a fraction and has the property that both its numerator and denominator are polynomials. In other words, R(x) is a rational function if R(x) = p(x) / q(x) where p(x) and q(x) are both polynomials. For example, The function R(x) = (x^2 + 4x - 1) / (3x^2 - 9x + 2) is a rational function since the numerator, x^2 + 4x - 1, is a polynomial and the denominator, 3x^2 - 9x + 2 is also a polynomial.

Case-defined function

Functions are represented by a combination of equations, each corresponding to a part of the domain. For example,

[pic 1]

One equation gives the values of ƒ(x) when x is less than

or equal to 1, and the other equation gives the values of ƒ(x) when x is greater than 1.

Absolute Value Function

In mathematics, the absolute value or modulus |x| of a real number x is the non-negative value of x without regard to its sign. Namely, |x| = x for a positive x, |x| = −x for a negative x, and |0| = 0. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3.

Factorials

Factorials are very simple things. They're just products, indicated by an exclamation mark. For instance, "fourfactorial" is written as "4!" and means 1×2×3×4 = 24. In general, n! ("enn factorial") means the product of all the whole numbers from 1 to n; that is, n! = 1×2×3×...×n.

Genetics

The probability of a function under certain conditions.

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