Fiche de révision : Understanding Relations and Functions

📋 Course Outline

1. Relations and Mappings
2. Arrow Diagram Representation
3. Ordered Pairs and Relations
4. Domain and Codomain
5. Range and Output Sets
6. Function Definition and Notation
7. Input and Output Variables
8. Functional Relationships
9. Multivariate and Bivariate Functions
10. Types of Functions

📖 1. Relations and Mappings

🔑 Key Concepts & Definitions

• Relations as mappings between two sets: In mathematics, relations are defined as connections between elements of two sets, where each element in the first set (domain) is related to one or more elements in the second set (codomain). These are often described as mappings, illustrating how elements from one set correspond to elements in another (source: Ms. Rukhsana Jabeen, 2026).

• Rule connecting pairs of elements in relations: A relation is characterized by a specific rule that establishes a connection between pairs of elements from the two sets. This rule must hold true for all pairs in the relation, defining the nature of the connection (source: Ms. Rukhsana Jabeen, 2026).

• Directional property of relations from one set to another: Relations possess a directional property, meaning the connection from an element in the first set (input or domain) to an element in the second set (output or codomain) is always oriented from the first to the second. This directionality is fundamental to the concept of relations as mappings (source: Ms. Rukhsana Jabeen, 2026).

📝 Essential Points

• Relations are studied as mappings that connect elements of two sets via a rule, often represented visually through arrow diagrams or as ordered pairs, with the direction always from the input set (domain) to the output set (codomain).
• The rule connecting pairs of elements must be consistent and applicable to all pairs in the relation, ensuring the relation's validity (source: Ms. Rukhsana Jabeen, 2026).
• Relations inherently have a directional property, emphasizing the flow from the domain to the codomain, which is crucial when defining functions and understanding their behavior (source: Ms. Rukhsana Jabeen, 2026).
• The concept of a relation involves three conditions: involvement of two sets, a clear rule, and a directional property from the domain to the codomain (source: Ms. Rukhsana Jabeen, 2026).

💡 Key Takeaway

Relations are structured connections between two sets, defined by a rule that establishes a directional mapping from elements of the first set to elements of the second, forming the foundation for understanding functions and their properties.

📖 2. Arrow Diagram Representation

🔑 Key Concepts & Definitions

• Arrow diagram: A visual tool used to represent relations between two sets, where members of each set are listed inside enclosed shapes, and arrows indicate the relation between related members (Ms. Rukhsana Jabeen, 2026).
• Members of sets listed inside enclosed shapes: The members of each set involved in a relation are written inside specific geometric shapes (such as circles or rectangles) in an arrow diagram, clearly identifying the elements involved (Ms. Rukhsana Jabeen, 2026).
• Arrows drawn to connect related members: Arrows are used to visually depict the relation by pointing from a member of the first set (input or domain) to a related member of the second set (output or codomain), illustrating the directional nature of the relation (Ms. Rukhsana Jabeen, 2026).

📝 Essential Points

• Arrow diagrams serve as an effective visual representation of relations between two sets, emphasizing the connection and directionality from members of the domain to members of the codomain (Ms. Rukhsana Jabeen, 2026).
• Members of each set are listed inside enclosed shapes, and arrows are drawn from each member of the first set to the related members of the second set, illustrating the relation clearly (Ms. Rukhsana Jabeen, 2026).
• The relation's rule must hold for all mappings, with arrows always directed from the input (domain) to the output (codomain), preserving the relation's directional property (Ms. Rukhsana Jabeen, 2026).
• Arrow diagrams can be complemented by ordered pairs, such as (4, 9), which preserve the directional property and align with Cartesian plane notation (Ms. Rukhsana Jabeen, 2026).

💡 Key Takeaway

Arrow diagrams provide a clear, visual method to represent relations between sets, using enclosed shapes for members and arrows to indicate the directional connections, thus illustrating the relation's structure effectively.

📖 3. Ordered Pairs and Relations

🔑 Key Concepts & Definitions

• Ordered pairs: A pair of elements written in a specific order, typically represented as (x, y), where the first element x is called the first component and the second element y is called the second component. These pairs are used to represent relations between two sets. (Source: Ms. Rukhsana Jabeen, 2026)

• Preservation of the directional property in ordered pairs: The ordered pair (x, y) maintains the directionality from the first component to the second, indicating a relation from x (input) to y (output). This directional property is essential for accurately representing relations and functions. (Source: Ms. Rukhsana Jabeen, 2026)

• Ordered pairs consistent with Cartesian plane notation: The notation (x, y) aligns with the Cartesian coordinate system, where x represents the horizontal position and y the vertical position. This consistency allows for graphical representation of relations on the Cartesian plane. (Source: Ms. Rukhsana Jabeen, 2026)

📝 Essential Points

• Relations between two sets can be represented using ordered pairs (x, y), which explicitly show the connection from an element in the first set (domain/input) to an element in the second set (codomain/output).
• The directional property in ordered pairs ensures the relation is meaningful; (x, y) indicates a relation from x to y, not vice versa. This is crucial for understanding mappings and functions.
• When representing relations graphically, ordered pairs correspond to points on the Cartesian plane, with the first component x indicating the horizontal position and the second component y indicating the vertical position.
• The notation (x, y) is fundamental in maintaining clarity and consistency in mathematical representations of relations, especially when analyzing their properties or visualizing them graphically.

💡 Key Takeaway

Ordered pairs (x, y) serve as the fundamental building blocks for representing relations, preserving the directional property from input to output, and aligning with Cartesian plane notation for graphical analysis.

📖 4. Domain and Codomain

🔑 Key Concepts & Definitions

• Domain: The set from which input values of a relation or function are taken. It specifies all possible values that can be input into the relation or function (see source content: "Relation between the set {4, 5, 6, 7} and the set {9, 10, 11, 12}... members of set X are the input").
• Codomain: The set onto which the outputs of a relation or function are mapped. It includes all potential output values that the relation or function could produce, regardless of whether they are actual outputs (see source content: "The entire set Y is called the codomain").
• Conditions for a relation involving domain and codomain:
  1. Two sets must be involved (the domain and the codomain).
  2. A clear rule describing the relationship between elements of the domain and codomain must exist.
  3. The relation must have a directional property, from the domain (inputs) to the codomain (possible outputs) (see source content: "relation is defined from one set called the domain onto another set called the codomain").

📝 Essential Points

• The domain is the set of all possible input values for a relation or function, serving as the source set for mappings.
• The codomain is the set that contains all potential output values; it is a target set onto which the outputs are mapped.
• The relation must involve two sets, a rule connecting elements, and a directional property from the domain to the codomain (see source content: "relation must hold for all mappings from set X to set Y").
• The range is a subset of the codomain, consisting of actual outputs produced by the relation or function (see source content: "The subset of Y consisting of the outputs is called the range").
• In functions, the domain and codomain are fundamental in defining the scope of inputs and potential outputs, respectively, with the rule ensuring each input maps to a unique output (see source content: "A function is a rule that assigns exactly one output to each input").

💡 Key Takeaway

The domain is the set of all possible inputs from which values are taken, while the codomain is the set of all potential outputs onto which these inputs are mapped, with the relation requiring a clear rule and directional property from the domain to the codomain.

📖 5. Range and Output Sets

🔑 Key Concepts & Definitions

• Range: The subset of the codomain that contains all actual outputs produced by a relation or function. It includes only those elements of the codomain that are mapped from the domain (see relation from set X to set Y).
  Pak-Austria Fachhochschule (2026): "The range is the set of all actual outputs of a relation or function."

• Codomain: The set into which all outputs of a relation or function are mapped; it is the target set defined at the outset, whether or not all elements are actually outputs.
  Pak-Austria Fachhochschule (2026): "The codomain is the set onto which the outputs are mapped, as specified in the relation or function."

• Difference between Range and Codomain: The range is a subset of the codomain containing only the elements that are actual outputs, whereas the codomain is the entire set designated as the possible outputs, regardless of whether they are realized.
  Pak-Austria Fachhochschule (2026): "Range is the set of actual outputs, while codomain is the set of all potential outputs."

📝 Essential Points

• A relation or function is defined with a domain and a codomain; the range is derived from the relation as the set of all outputs actually produced from the domain inputs.
• The range is always a subset of the codomain. Not all elements of the codomain need to be outputs; some may never be mapped to by any element of the domain.
• For example, if a relation from set $X = \{1, 2, 3, 4\}$ to set $Y = \{2, 3, 4, 5, 6, 7, 8\}$ is defined by $x \to 2x$, then the range is $\{2, 4, 6, 8\}$, while the codomain remains $\{2, 3, 4, 5, 6, 7, 8\}$. Elements like 3, 5, and 7 are in the codomain but not in the range because they are not outputs of the relation.
• Understanding the distinction helps clarify what outputs are actually possible versus what are merely potential, as set by the codomain.

💡 Key Takeaway

The range of a relation or function is the set of all actual outputs produced, which is always a subset of the codomain—the set of all possible outputs specified at the outset. Recognizing this distinction is essential for analyzing the behavior and characteristics of relations and functions.

📖 6. Function Definition and Notation

🔑 Key Concepts & Definitions

• Function (general): A mathematical rule that assigns exactly one output value to each input value. This ensures that for every input, there is a unique corresponding output, establishing a clear relationship between the two. (Source: Ms. Rukhsana Jabeen, 2026)

• Notation of a function: The process of representing the relationship between input and output using symbols, typically written as f(x) = y, where f indicates the rule, x is the input, and y is the output. This notation succinctly describes the functional relationship. (Source: Ms. Rukhsana Jabeen, 2026)

• Mapping notation: A way to represent the assignment of inputs to outputs explicitly, often written as f(x) = y. It emphasizes the process of mapping each input x to a specific output y via the rule f. (Source: Ms. Rukhsana Jabeen, 2026)

📝 Essential Points

• A function differs from a general relation because it guarantees one and only one output for each input, which is critical for defining well-structured mathematical relationships (see section 1 for relations).

• The notation f(x) = y is fundamental in expressing functions, where f is the rule, x is the input variable (independent variable), and y is the output variable (dependent variable). This notation helps identify the input-output relationship clearly.

• The mapping notation emphasizes the process of assigning each input x to a specific output y through the rule f, illustrating the functional relationship visually and algebraically.

• Functions can be multivariate (more than one input variable), denoted as z = f(x_1, x_2, ..., x_n), where the output depends on multiple independent variables.

• The domain of a function is the set of all possible inputs, while the range is the set of all actual outputs corresponding to the inputs.

💡 Key Takeaway

A function is a precise rule that assigns exactly one output to each input, represented by the notation f(x) = y and visualized through mapping diagrams, forming the foundation for understanding relationships between variables in mathematics.

📖 7. Input and Output Variables

🔑 Key Concepts & Definitions

• Input variables (independent variables): Variables that serve as the starting point in a function, whose values are chosen freely or are given, and from which the output depends. They are the "inputs" in the functional relationship. (Source: Ms. Rukhsana Jabeen, 2026)

• Output variables (dependent variables): Variables that depend on the input variables within a function. Their values are determined by the rule or relationship applied to the input variables. They are the "outputs" generated from the inputs. (Source: Ms. Rukhsana Jabeen, 2026)

• Directional flow in functions: The relationship in a function flows from input variables to output variables, indicating that the output depends on the input, but not vice versa. This unidirectional flow is fundamental to the concept of functions. (Source: Ms. Rukhsana Jabeen, 2026)

📝 Essential Points

• In functions, input variables are termed independent variables because their values are not affected by other variables within the function, and they can be freely chosen or given. (Source: Ms. Rukhsana Jabeen, 2026)

• Output variables are dependent variables because their values depend on the input variables through the rule or relationship defined by the function. The output is uniquely determined once the input is known. (Source: Ms. Rukhsana Jabeen, 2026)

• The directional flow from input to output reflects the core property of functions: the output variable is a function of the input variable(s). This ensures the relationship is well-defined and unidirectional, which is critical in mathematical modeling and real-world applications. (Source: Ms. Rukhsana Jabeen, 2026)

• The notation f(x) = y exemplifies the relationship where x is the input (independent variable), and y is the output (dependent variable). The rule f maps each input x to a unique output y. (Source: Ms. Rukhsana Jabeen, 2026)

💡 Key Takeaway

Input variables are the independent factors that determine the output in a function, with the flow of influence always moving from input to output, ensuring a clear, unidirectional relationship.

📖 8. Functional Relationships

🔑 Key Concepts & Definitions

• Functional relationship expressed as y = f(x): A mathematical expression where the output variable y is defined explicitly as a function of the input variable x, indicating that y depends on x in a specific, rule-based manner (see source content).
• Dependence of output variable on input variable: The principle that the value of the output (dependent variable) is determined by the input (independent variable) through a specific rule or relationship, emphasizing the directional nature of the relationship (see source content).
• Examples of functional relationships in real-world contexts: Practical instances where the output depends on an input, such as taxi fares depending on distance, house prices depending on location, or fees based on program type, illustrating the application of functions outside pure mathematics (see source content).

📝 Essential Points

• A functional relationship is often expressed as y = f(x), where y is the dependent variable and x is the independent variable, indicating that y's value is uniquely determined by x (see source).
• The dependence of the output on the input is fundamental to functions, meaning for each input x, there is exactly one output y, establishing a one-to-one correspondence in the context of the rule f (see source).
• Real-world examples, such as transportation costs, real estate prices, and educational fees, demonstrate how functions model practical relationships where the output varies predictably with the input (see source).
• The functional relationship is characterized by a rule that assigns each input exactly one output, ensuring the relationship's consistency and predictability, which is essential for mathematical modeling and analysis (see source).
• Understanding the dependence of variables helps in analyzing how changes in input influence the output, crucial for decision-making and predictions in various fields (see source).

💡 Key Takeaway

A functional relationship expressed as y = f(x) captures how an output depends on an input through a specific rule, enabling the modeling of real-world scenarios where one variable uniquely determines another.

📖 9. Multivariate and Bivariate Functions

🔑 Key Concepts & Definitions

• Multivariate functions: Functions that involve more than one independent variable, where the dependent variable is expressed as a function of multiple inputs. (Rukhsana Jabeen, 2026): "A function which contains more than one independent variable are called multivariate functions."
• Bivariate functions: A specific type of multivariate function involving exactly two independent variables. (Rukhsana Jabeen, 2026): "A function having two independent variables is called bivariate function."
• Notation of multivariate functions: These functions are generally expressed as $z = f(x_1, x_2, ..., x_n)$, where $x_1, x_2, ..., x_n$ are the independent variables, and $z$ is the dependent variable. (Rukhsana Jabeen, 2026): "In general the notation for a function $f$ where the value of dependent variable depends on the values of $n$ independent variables is $z = f(x_1, x_2, ..., x_n)$."

📝 Essential Points

• Multivariate functions extend the concept of functions beyond a single input, allowing modeling of complex relationships involving multiple factors.
• Bivariate functions are a common subset, often used in applications like economics, physics, and engineering, where two independent variables influence an outcome.
• The notation $z = f(x_1, x_2, ..., x_n)$ clearly indicates the dependence of the dependent variable $z$ on multiple independent variables, facilitating understanding of multi-factor relationships.
• These functions are crucial for analyzing systems where multiple inputs collectively determine an output, aligning with the broader study of relations and functions (see section 6).

💡 Key Takeaway

Multivariate and bivariate functions generalize the concept of single-variable functions to multiple inputs, enabling the analysis of complex, real-world relationships involving several independent factors.

📖 10. Types of Functions

🔑 Key Concepts & Definitions

• Classification of functions: The process of categorizing functions based on their properties, such as the nature of their rule, the number of variables involved, or their behavior (e.g., linear, quadratic, polynomial, exponential, logarithmic). Ms. Rukhsana Jabeen (2026) emphasizes that understanding these classifications helps in analyzing the behavior and applications of different functions.

• Examples of function types:

  • Linear functions: Functions where the rule is a first-degree polynomial, expressed as $f(x) = mx + c$, with a constant rate of change.
  • Quadratic functions: Functions with a second-degree polynomial rule, $f(x) = ax^2 + bx + c$, characterized by a parabolic graph.
  • Polynomial functions: Functions involving sums of powers of $x$, such as $f(x) = a_nx^n + \dots + a_1x + a_0$.
  • Exponential functions: Functions where the variable appears as an exponent, $f(x) = a^x$, with applications in growth and decay.
  • Logarithmic functions: The inverse of exponential functions, expressed as $f(x) = \log_a x$.

• Characteristics distinguishing types of functions:

  • Domain and Range: Different functions have specific domains and ranges; for example, exponential functions have a domain of all real numbers and a range of positive real numbers.
  • Graph shape: Linear functions produce straight lines, quadratic functions produce parabolas, and exponential functions produce curves that grow or decay rapidly.
  • Behavior: Polynomial functions can have multiple turning points, while exponential functions exhibit continuous growth or decay.

📝 Essential Points

• Classification of functions is based on their algebraic form, graph shape, and behavior (Ms. Rukhsana Jabeen, 2026).
• Examples of common function types include linear, quadratic, polynomial, exponential, and logarithmic functions, each with unique characteristics.
• Characteristics such as the degree of the polynomial, the nature of the rule (e.g., constant, increasing, decreasing), and the shape of the graph help distinguish between different types.
• Multivariate functions involve more than one independent variable, such as $z = f(x, y)$, and are classified accordingly (see section 9).

💡 Key Takeaway

Functions are classified into various types based on their algebraic form, graph shape, and behavior, which aids in understanding their properties and applications in different contexts. Recognizing these types allows for better analysis and problem-solving in mathematics.

📊 Synthesis Tables

⚠️ Common Pitfalls & Confusions

1. Confusing relations with functions; relations may assign multiple outputs to a single input, while functions cannot.
2. Misinterpreting the directionality in ordered pairs; (x, y) indicates a relation from x to y, not vice versa.
3. Overlooking the difference between domain (inputs) and codomain (potential outputs); the codomain may include elements not actually mapped.
4. Assuming all relations are functions; only those with exactly one output per input qualify as functions.
5. Misrepresenting arrow diagrams—failing to draw arrows from domain to codomain, or mixing set members.
6. Forgetting that the range is the subset of the codomain actually mapped to by the relation or function.
7. Confusing the codomain with the range; the range is the set of actual outputs, which is a subset of the codomain.

✅ Exam Checklist

• Know the definition of relations as mappings between two sets, including the rule connecting pairs (Ms. Rukhsana Jabeen, 2026).
• Be able to represent relations using arrow diagrams and ordered pairs, maintaining the directionality from domain to codomain.
• Understand the concept of ordered pairs and their role in relation representation, aligning with Cartesian notation.
• Distinguish between domain (input set) and codomain (potential output set), and explain their roles in relations and functions.
• Define a function and differentiate it from a general relation; recall that each input has exactly one output.
• Recognize multivariate and bivariate functions, and understand their differences.
• Identify various types of functions: one-to-one, onto, many-to-one, constant, linear, quadratic, with their properties.
• Know SMITH's definition of the "invisible hand" in economic context, if applicable.
• Be familiar with arrow diagram conventions and how to interpret them.
• Understand the significance of the range as the actual set of outputs, and how it differs from the codomain.
• Recall the conditions for a relation involving two sets, a rule, and directionality.
• Be able to analyze and classify functions based on their properties and representations.