MATEMATIKA REKAYASA IMO 141202

Kuliah 2: Fungsi & Limt

mahmud mustain

MATERI Kuliah 2

Fungsi dan Limits: 1. Domain dan range fungsi, 2. Fungsi linear, 3. Fungsi kuadratik dan trigonometri, 4. Fungsi hyperbolik, 5. Limits dan kontinuitas fungsi.

Definisi(Wikipedia)

• 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.

Fungsi Linear

In calculus and related areas, a linear function is a polynomial function of degree zero or one, or is the zero polynomial.

In linear algebra and functional analysis, a linear function is a linear map.

In linear algebra, a linear function is a map f between two vector spaces that preserves vector addition and scalar multiplication:

Here a denotes a constant belonging to some field K of scalars (for example, the real numbers) and x and y are elements of a vector space, which might be K itself.

Some authors use "linear function" only for linear maps that take values in the scalar field;[4] these are also called linear functionals.

Quadratic function• A univariate quadratic function, in mathematics, is a polynomial function of

the form of, • in a single variable x. The graph of a univariate 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 univariate 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 function.

• A quadratic function can also be multivariate (having more than one variable). The bivariate case in terms of variables x and y has the form of

• In general there can be an arbitrarily large number of variables, but the highest degree term must be of degree 2, such as x2, xy, yz, etc.

Forms of a quadratic function• The standard form,

• The factored form, where x1 and x2 are the roots of the quadratic equation, it is used in logistic map

• The vertex form, where h and k are the x and y coordinates of the vertex, respectively.

Trigonometric functions

• In mathematics, the trigonometric functions (also called the circular functions) are functions of an angle. They relate the angles of a triangle to the lengths of its sides. Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications.

Hyperbolic function

• In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions.

The inverse hyperbolic functions are the area hyperbolic sine "arsinh" (also called "asinh" or sometimes "arcsinh")[2] and so on.

Standard algebraic expressions

Limit of a function

• In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input.

• Formal definitions, first devised in the early 19th century, are given below. Informally, a function f assigns an output f(x) to every input x. We say the function has a limit L at an input p: this means f(x) gets closer and closer to L as x moves closer and closer to p. More specifically, when f is applied to any input sufficiently close to p, the output value is forced arbitrarily close to L. On the other hand, if some inputs very close to p are taken to outputs that stay a fixed distance apart, we say the limit does not exist.

Continuous function

• In mathematics, a continuous function is, roughly speaking, a function for which small changes in the input result in small changes in the output. Otherwise, a function is said to be a discontinuous function. A continuous function with a continuous inverse function is called a homeomorphism.

• As an example, consider the function h(t), which describes the height of a growing flower at time t. This function is continuous. By contrast, if M(t) denotes the amount of money in a bank account at time t, then the function jumps whenever money is deposited or withdrawn, so the function M(t) is discontinuous.

• Definition in terms of limits of functions• The function f is continuous at some point c of

its domain if the limit of f(x) as x approaches c through the domain of f exists and is equal to f(c).[2]

• In mathematical notation, this is written as

Definition using oscillation• The failure of a function to be

continuous at a point is quantified by its oscillation.

• Continuity can also be defined in terms of oscillation: a function f is continuous at a point x0 if and only if its oscillation at that point is zero;[3] in symbols,

• A benefit of this definition is that it quantifies discontinuity: the oscillation gives how much the function is discontinuous at a point.

Tugas 1a

Tugas 1b

AL-HAMDULILLAH