Function

BASIC DEFINITION

A function is a mathematical relation that assigns exactly one element from a set of values (called the codomain or range) to each element from a set of definitions (also called the domain). Symbolically written:

f : D → Z, where for every x ∈ D, there exists exactly one y ∈ Z such that f(x) = y.

Thus, a function describes the dependency between two quantities: x (the independent variable) and y (the dependent variable).

REPRESENTATION OF A FUNCTION

A function can be given in various ways:

• by a formula (e.g., f(x) = x^2 + 2),
• tabularly (a list of values),
• graphically (the graph of the function in a coordinate system),
• descriptively (with a textual description of the connection between quantities).

DOMAIN AND RANGE (SET OF VALUES)

Domain (notation Df): all permissible x for which the function is defined.
Range (Set of Values) (notation Zf): all y that the function actually achieves.

For example:
f(x) = sqrt(x) is defined only for x >= 0, so Df = [0, infinity).
Zf = [0, infinity), because the square root does not return negative numbers.

TYPES OF FUNCTIONS

Functions can be classified according to their form and characteristics:

• linear function: f(x) = kx + n,
• quadratic function: f(x) = ax^2 + bx + c,
• exponential function: f(x) = a^x,
• logarithmic function: f(x) = log_a(x),
• rational function: f(x) = p(x)/q(x), where p and q are polynomials.

PROPERTIES OF FUNCTIONS

Functions are also analyzed based on:

• monotonicity (increasing/decreasing),
• parity (odd, even functions),
• continuity,
• periodicity,
• limits,
• derivatives (in higher mathematics).

EXAMPLE

Let's define the function f(x) = x^2 – 2x + 1.
This is a quadratic function, and its graph is a parabola.
Df = R (all real numbers), because it is defined for all real numbers.
Zf = [0, infinity), because it has a minimum at x = 1, where f(1) = 0.

CONCLUSION

A function is a fundamental concept in mathematics that allows for the description of dependencies between quantities. Through various representations and analyses of functions, we explore their behavior and properties, which is crucial in all further chapters of algebra, analysis, and applied mathematics. explore their behavior and properties, which is crucial in all further chapters of algebra, analysis, and applied mathematics.