Fungsi matematika

Fungsi, dalam matematika, ekspresi, aturan, atau hukum yang mendefinisikan hubungan antara satu variabel (variabel independen) dan variabel lain (variabel dependen). Fungsi ada di mana-mana dalam matematika dan sangat penting untuk merumuskan hubungan fisik dalam sains. Definisi fungsi modern pertama kali diberikan pada tahun 1837 oleh ahli matematika Jerman Peter Dirichlet:

analisis: Fungsi

Kalkulus memperkenalkan ahli matematika ke banyak fungsi baru dengan menyediakan cara baru untuk mendefinisikannya, seperti dengan deret tak hingga dan dengan integral.

Jika variabel y sangat terkait dengan variabel x sehingga setiap kali nilai numerik diberikan ke x, ada aturan yang sesuai dengan nilai unik y ditentukan, maka y dikatakan sebagai fungsi dari variabel independen x.

Hubungan ini umumnya dilambangkan sebagai y = f (x). Selain f (x), simbol disingkat lainnya seperti g (x) dan P (x) sering digunakan untuk mewakili fungsi variabel independen x, terutama ketika sifat fungsi tidak diketahui atau tidak ditentukan.

Fungsi umum

Banyak rumus matematika yang banyak digunakan adalah ekspresi dari fungsi yang diketahui. Misalnya, rumus untuk area lingkaran, A = πr ², memberikan variabel dependen A (area) sebagai fungsi dari variabel independen r (jari-jari). Fungsi yang melibatkan lebih dari dua variabel juga umum dalam matematika, seperti yang dapat dilihat dalam rumus untuk luas segitiga, A = bh / 2, yang mendefinisikan A sebagai fungsi dari b (basis) dan h (tinggi). Dalam contoh-contoh ini, kendala fisik memaksa variabel independen menjadi bilangan positif. Ketika variabel independen juga diizinkan untuk mengambil nilai negatif — dengan demikian, bilangan real apa pun — fungsinya dikenal sebagai fungsi bernilai riil.

Rumus untuk luas lingkaran adalah contoh fungsi polinomial. Bentuk umum untuk fungsi tersebut adalah P (x) = a ₀ + a ₁ x + a ₂ x ² + ⋯ + a _n x ⁿ, di mana koefisien (a ₀, a ₁, a ₂,

, a _n) diberikan, x dapat berupa bilangan real, dan semua kekuatan x menghitung bilangan (1, 2, 3,

). (When the powers of x can be any real number, the result is known as an algebraic function.) Polynomial functions have been studied since the earliest times because of their versatility—practically any relationship involving real numbers can be closely approximated by a polynomial function. Polynomial functions are characterized by the highest power of the independent variable. Special names are commonly used for such powers from one to five—linear, quadratic, cubic, quartic, and quintic.

Polynomial functions may be given geometric representation by means of analytic geometry. The independent variable x is plotted along the x-axis (a horizontal line), and the dependent variable y is plotted along the y-axis (a vertical line). The graph of the function then consists of the points with coordinates (x, y) where y = f(x). For example, the graph of the cubic equation f(x) = x³ − 3x + 2 is shown in the figure.

Another common type of function that has been studied since antiquity is the trigonometric functions, such as sin x and cos x, where x is the measure of an angle (see figure). Because of their periodic nature, trigonometric functions are often used to model behaviour that repeats, or “cycles.” Nonalgebraic functions, such as exponential and trigonometric functions, are also known as transcendental functions.

Complex functions

Practical applications of functions whose variables are complex numbers are not so easy to illustrate, but they are nevertheless very extensive. They occur, for example, in electrical engineering and aerodynamics. If the complex variable is represented in the form z = x + iy, where i is the imaginary unit (the square root of −1) and x and y are real variables (see figure), it is possible to split the complex function into real and imaginary parts: f(z) = P(x, y) + iQ(x, y).

Inverse functions

By interchanging the roles of the independent and dependent variables in a given function, one can obtain an inverse function. Inverse functions do what their name implies: they undo the action of a function to return a variable to its original state. Thus, if for a given function f(x) there exists a function g(y) such that g(f(x)) = x and f(g(y)) = y, then g is called the inverse function of f and given the notation f⁻¹, where by convention the variables are interchanged. For example, the function f(x) = 2x has the inverse function f⁻¹(x) = x/2.

Other functional expressions

A function may be defined by means of a power series. For example, the infinite series

could be used to define these functions for all complex values of x. Other types of series and also infinite products may be used when convenient. An important case is the Fourier series, expressing a function in terms of sines and cosines:

Such representations are of great importance in physics, particularly in the study of wave motion and other oscillatory phenomena.

Sometimes functions are most conveniently defined by means of differential equations. For example, y = sin x is the solution of the differential equation d²y/dx² + y = 0 having y = 0, dy/dx = 1 when x = 0; y = cos x is the solution of the same equation having y = 1, dy/dx = 0 when x = 0.