Function (mathematics)From Wikipedia, the free encyclopedia

A function f takes an input x, and returns a single output f(x). One metaphor describes the function as a "machine" or "black box" that for each input returns a corresponding output.

The red curve is the graph of a function f in the Cartesian plane, consisting of all points with coordinates of the form (x,f(x)). The property of having one output for each input is represented geometrically by the fact that each vertical line (such as the yellow line through the origin) has exactly one crossing point with the curve.

Function

x ↦ f(x)

By domain and codomain

■ —› B Bn —› B

■ —› Z —› ■

■ —› R —› ■ Rn —› ■

■ —› C —› ■ Cn —› ■

 Classes/properties  Constant · Identity · Linear · Polynomial · Rational ·

Algebraic · Analytic · Smooth · Continuous · Measurable

  Constructions  Restriction · Composition · λ · Inverse

  Generalizations  Partial · Multivalued · Implicit

v t e

In mathematics, a function[1] is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x2. The output of a function f corresponding to an input x is denoted by f(x) (read "f of x"). In this example, if the input is −3, then the output is 9, and we may write f(−3) = 9. The input variable(s) are sometimes referred to as the argument(s) of the function.

Functions of various kinds are "the central objects of investigation"[2] in most fields of modern mathematics. There are many ways to describe or represent a function. Some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. Others

are given by a picture, called the graph of the function. In science, functions are sometimes defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation.

The input and output of a function can be expressed as an ordered pair, ordered so that the first element is the input (or tuple of inputs, if the function takes more than one input), and the second is the output. In the example above, f(x) = x2, we have the ordered pair (−3, 9). If both input and output are real numbers, this ordered pair can be viewed as the Cartesian coordinates of a point on the graph of the function. But no picture can exactly define every point in an infinite set.

In modern mathematics,[3] a function is defined by its set of inputs, called the domain; a set containing the set of outputs, and possibly additional elements, as members, called its codomain; and the set of all input-output pairs, called its graph. (Sometimes the codomain is called the function's "range", but warning: the word "range" is sometimes used to mean, instead, specifically the set of outputs. An unambiguous word for the latter meaning is the function's "image". To avoid ambiguity, the words "codomain" and "image" are the preferred language for their concepts.) For example, we could define a function using the rule f(x) = x2 by saying that the domain and codomain are the real numbers, and that the graph consists of all pairs of real numbers (x, x2). Collections of functions with the same domain and the same codomain are called function spaces, the properties of which are studied in such mathematical disciplines as real analysis, complex analysis, and functional analysis.

In analogy with arithmetic, it is possible to define addition, subtraction, multiplication, and division of functions, in those cases where the output is a number. Another important operation defined on functions is function composition, where the output from one function becomes the input to another function.

ContentsThe linear function is popular in economics. It is attractive because it is simple and easy to handle mathematically. It has many important applications.

Linear functions are those whose graph is a straight line.

A linear function has the following form

y = f(x) = a + bx

A linear function has one independent variable and one dependent variable. The independent variable is x and the dependent variable is y.

a is the constant term or the y intercept. It is the value of the dependent variable when x = 0.

b is the coefficient of the independent variable. It is also known as the slope and gives the rate of change of the dependent variable.

 

Graphing a linear function

To graph a linear function:

1. Find 2 points which satisfy the equation

2. Plot them

3. Connect the points with a straight line

Example:

y = 25 + 5x

let x = 1then y = 25 + 5(1) = 30

let x = 3then y = 25 + 5(3) = 40

 

 

A simple example of a linear equation

A company has fixed costs of $7,000 for plant and equuipment and variable costs of $600 for each unit of output. What is total cost at varying levels of output?

let x = units of outputlet C = total cost

C = fixed cost plus variable cost = 7,000 + 600 x

output total cost15 units C = 7,000 + 15(600) = 16,00030 units C = 7,000 + 30(600) = 25,000

 

 

Combinations of linear equations

Linear equations can be added together, multiplied or divided.

A simple example of addition of linear equations

C(x) is a cost function

C(x) = fixed cost + variable cost

R(x) is a revenue function

R(x) = selling price (number of items sold)

profit equals revenue less cost

P(x) is a profit function

P(x) = R(x) - C(x)

x = the number of items produced and sold

 

Data:

A company receives $45 for each unit of output sold. It has a variable cost of $25 per item and a fixed cost of $1600. What is its profit if it sells (a) 75 items, (b)150 items, and (c) 200 items?

 

R(x) = 45x C(x) = 1600 + 25x

P(x) = 45x -(1600 + 25x)  

       = 20x - 1600  

let x = 75 P(75) = 20(75) - 1600 = -100        a loss

let x = 150 P(150) = 20(150) - 1600 = 1400let x = 200 P(200) = 20(200) - 1600 = 2400

A quadratic function, in mathematics, is a polynomial function of the form

[1]

The graph of a quadratic function is a parabola whose axis of symmetry is parallel to the y-axis.

The expression ax2 + bx + c in the definition of a quadratic function is a polynomial of degree 2, or a 2nd degree polynomial, because the highest exponent of x is 2. This expression is also called a quadratic polynomial or quadratic.

If the quadratic function is set equal to zero, then the result is a quadratic equation. The solutions to the equation are called the roots of the equation.