Mathematics - Types of Relations and Functions

Study Notes

Types of Relations

Relation: A set of ordered pairs (x, y).
Types:
- One-to-One: Each input has a unique output; no two distinct inputs share the same output.
- Many-to-One: Multiple inputs can map to the same output.
- Onto (Surjective): Every element in the codomain has at least one pre-image in the domain.
- One-to-Many: A single input maps to multiple outputs (not a function).
- Many-to-Many: Multiple inputs map to multiple outputs (not a function).

Function Properties

Definition: A relation where each input is related to exactly one output.
Notation: f(x) denotes a function f applied to x.
Properties:
- Injective: No two distinct inputs map to the same output.
- Surjective: Covers the entire codomain.
- Bijective: Both injective and surjective; each input has a unique output and covers the codomain.

Domain and Range

Domain: The set of all possible inputs (x-values) for a function.
Range: The set of all possible outputs (y-values) that the function can produce.
Finding Domain:
- Identify values that make the function undefined (e.g., division by zero).
Finding Range:
- Determine possible output values based on the function’s behavior.

Inverse Functions

Definition: A function that reverses the effect of the original function.
Notation: If f(x) = y, then f⁻¹(y) = x.
Finding Inverse:
1. Replace f(x) with y.
2. Solve for x in terms of y.
3. Swap x and y to get f⁻¹(x).
Conditions: A function must be bijective to have an inverse (one-to-one correspondence).

Composition of Functions

Definition: Combining two functions where the output of one function becomes the input of another.
Notation: (f ∘ g)(x) = f(g(x)).
Properties:
- Not generally commutative: f(g(x)) ≠ g(f(x)).
- Associative: f(g(h(x))) = (f ∘ g ∘ h)(x).
Finding Composition:
- Substitute the inner function into the outer function.

Types of Relations

A relation consists of a set of ordered pairs (x, y).
One-to-One: Each input corresponds to a unique output; no outputs are shared among different inputs.
Many-to-One: Multiple inputs can produce the same output.
Onto (Surjective): Every element in the codomain has at least one pre-image in the domain.
One-to-Many: A single input can relate to multiple outputs; this does not qualify as a function.
Many-to-Many: Multiple inputs can relate to multiple outputs; this is also not a function.

Function Properties

A function relates each input to exactly one output.
Notation: f(x) signifies the function f evaluated at x.
Injective: Distinct inputs are mapped to distinct outputs, ensuring no overlap.
Surjective: The function's outputs encompass every element in the codomain.
Bijective: Combines properties of injective and surjective; each input has a unique output, and every codomain element is covered.

Domain and Range

Domain: Represents all possible inputs (x-values) for the function.
Range: Captures all possible outputs (y-values) generated by the function.
Finding the domain involves identifying values that result in undefined outputs, such as division by zero.
Determining the range requires analyzing the function's behavior to ascertain attainable y-values.

Inverse Functions

An inverse function undoes the mapping of the original function.
Notation: If f(x) yields y, then the inverse is represented as f⁻¹(y) = x.
Finding the Inverse:
- Replace f(x) with y.
- Solve the equation for x in terms of y.
- Swap x and y to express the inverse, f⁻¹(x).
A function must be bijective to possess an inverse, ensuring a one-to-one correspondence between inputs and outputs.

Composition of Functions

Composition involves joining two functions, where the output of one becomes the input for the other.
Notation: (f ∘ g)(x) indicates the composition where g is evaluated first, followed by f.
Composition is generally not commutative; f(g(x)) does not equal g(f(x)).
Composition is associative; f(g(h(x))) simplifies to (f ∘ g ∘ h)(x).
To perform composition, substitute the resultant output of the inner function into the outer function.

Description

Dive into the world of relations and functions in this quiz. Explore various types of relations such as one-to-one, many-to-one, and more. Test your knowledge on function properties including injective, surjective, and bijective characteristics.