Mathematical Functions: Definitions, Types, and Properties

Mathematical Functions: Core Concepts

Function: A rule or correspondence that associates each element $x$ of one set with one and only one element of a set $y$. It is a special relationship between elements of two sets.

Elements of a Function

• Domain: For a function $f: A \to B$, the domain corresponds to the first set ($A$).
• Codomain: The second set ($B$), which contains all possible values related to the elements of the domain.
• Image of a Function: The set of all output values obtained by applying the function $f$ to the elements of the domain $A$. If $f: A \to B$, the image is a subset of $B$. This implies a partnership where each element of set $A$ maps to a single element of set $B$.

Characteristics of a Function

A function is a relationship between two sets, not necessarily involving numbers, where each element in the first set corresponds to exactly one element in the second set.

Correspondence Rule: This establishes a relationship that associates each element in the first set with exactly one other element in the second set.

Intervals

The length comprising from one point to another on the $xy$-axis, having two limits (upper and lower).

• Open Intervals: Do not consider the limits as actual values of the interval (denoted by an open circle or parentheses).
• Closed Intervals: Consider their limits as included values and are denoted by a closed circle or square brackets.

Inequalities

An algebraic expression whose solution set is determined by an infinite number of values. Inequalities are solved using rules similar to equality, with fundamental changes, such as reversing inequality signs when multiplying or dividing both sides by a negative number.

Types of Functions

Functions are broadly categorized as algebraic and transcendental, continuous and discontinuous, increasing and decreasing.

Algebraic Functions

These are functions that can be constructed using algebraic operations: addition, subtraction, multiplication, division, and root extraction from polynomials.

Transcendental Functions

A function that contains a variable affecting the root or is used in exponential and logarithmic forms.

Exponential Functions

Those whose base is a fixed number and the exponent is a variable. There are two main types: those based on any positive whole number, and those whose base is the natural logarithm ($e \approx 2.7182818$).

Properties of Exponential Functions

Governed by exponent laws, which extend to rational exponents (e.g., $a^{m/n} = \sqrt[n]{a^m}$, where $m$ and $n$ are non-zero whole numbers, and $a^{0} = 1$). The graphs of exponential functions are typically of two types: ascending or descending.

Logarithmic Functions

These are primarily used to simplify arithmetic operations.

Continuity and Discontinuity

A function is continuous at a number $a$ if $\lim_{x \to a} f(x) = f(a)$. It is discontinuous at $a$ if it is not continuous at $a$.

Increasing and Decreasing Functions

An interval is increasing if for any $x_1 < x_2$ in the interval, $f(x_1) < f(x_2)$. Conversely, it is decreasing if $f(x_1) > f(x_2)$ when $x_1 < x_2$.

Inverse Function

The reciprocal of a function, denoted by $f^{-1}(x)$, which reverses the action performed by $f(x)$. Finding the inverse often requires isolating the independent variable.

Special Function Types

Different functions are characterized by unique algorithms:

• Absolute Value Function: Defined by two vertical lines; the output value is always non-negative.
• Step Function: Formed by two or more horizontal segments (steps) over defined intervals on the $x$-axis.
• Polynomial Functions: Algebraic functions whose rule is a polynomial of a certain degree.
• Polynomial: A structure containing more than three terms.
• Linear Function: The expression or function whose equation is $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept.
• Constant Function: If its equation is $f(x) = a$, where $a$ is a constant value in the domain of real numbers ($\mathbb{R}$).
• Quadratic Function: It is of the form $f(x) = ax^2 + bx + c$, where $x$ is the variable.