Mathematical Functions: Domain, Range, and Properties

Understanding Mathematical Functions

A relationship based on two numerical variables, usually called X and Y. X is the independent variable, and Y is the dependent variable. The function is usually denoted by y = f(x), associating each value of x with a single value of y: x → y = f(x).

Domain and Image of a Function

Domain: The domain of a function f(x) consists of all values of X for which the function is defined.

Image (Range): Im(f) is the set of values that the function takes. That is, the set of values y for which there exists an x such that f(x) = y.

Methods of Representing Functions

Ways to represent functions:

• Through its graphical expression
• Through a statement
• By a table of values
• By an analytical expression or formula

Calculating Different Types of Domains

• Denominators: The values that make a denominator zero are not in the domain of definition. For example, for f(x) = 1 / (x + 3), the domain is the set of all real numbers except x = -3. I.e., (-∞, -3) ∪ (-3, +∞).
• Square roots: Values that make the expression under the root negative are not in the domain. For example, in f(x) = √(x - 2), the values x < 2 are not in the domain of definition. So, Dom(f) = [2, +∞).
• Polynomials: A function like x² + 3x - 2 is defined for all real numbers and is resolved as a normal equation.

Continuity and Monotonicity

Continuous function: A function with no discontinuities of any kind. A function is continuous on an interval (a, b) if there is no discontinuity within it.

Increasing Function (Crescent): If x₁ < x₂, then f(x₁) < f(x₂).

Decreasing Function: If x₁ < x₂, then f(x₁) > f(x₂).

Relative Extrema and Periodicity

A function has a relative maximum at a point when the function takes a value greater than the surrounding points. In this case, the function is increasing to the maximum and decreasing after it. Analogously, a relative minimum occurs if the function is decreasing before the point and increasing after it.

Trend and Periodicity: There are functions where, even if we only know a piece of them, we can predict how they will behave far from the studied range because they have branches with a very clear trend.

Periodicity: A periodic function is one whose behavior is repeated each time the independent variable crosses a certain range. The length of this interval is called a period.