Lecture on Functions

Introduction to Functions

• Functions have been covered in mathematics, but are explored more deeply in pure mathematics.
• Definition: Think of a function as a machine with an input and an output.
  • Example: Input 5, machine multiplies by 5 to give output 25.
• A function defines a relationship between two variables, an independent and a dependent.
• A function relates an element of a set to exactly one element of another set.

Representations of Functions

• Functions can be represented in various ways:
  • Formula
  • Graph
  • List of values
  • Set of ordered pairs
  • Arrow diagram (mapping)

Function Notation

• Represented as: ( y = f(x) )
  • Example: ( f(x) = 2x + 1 )
• For every x value, there is a unique y value.

Fundamental Concepts

Ordered Pairs

• Comprised of an x coordinate and a y coordinate (e.g., (1, 2)).
• Used to plot graphs.

Relations

• A relation is a set of ordered pairs.
• Not all relations are functions, but all functions are relations.
• Types of Relations: One-to-One, Many-to-One (functions), One-to-Many, Many-to-Many (not functions).

Domain, Codomain, and Range

• Domain: Set of input values.
• Codomain: Set of potential output values.
• Range: Actual output values from the function.
  • Distinction: Range is a subset of the codomain.

Mapping Relations

• Represented using diagrams called mappings.
• X-coordinates are elements of the domain; Y-coordinates are elements of the codomain.
• Only elements connected by arrows from domain to codomain are part of the range.

Testing for Functions

Vertical Line Test

• Determines if a relation is a function.
• If a vertical line intersects the graph at more than one point, it's not a function.

Types of Functions

Injective (One-to-One)

• Every element of the domain maps to a unique element in the codomain.
• Algebraic Method: Prove that ( a = b ) when ( f(a) = f(b) ).
• Horizontal Line Test: Line drawn parallel to x-axis should intersect only once.

Surjective (Onto)

• Every element of the codomain is mapped by some element of the domain.
• Range is equal to the codomain.
• Algebraic Method: Show for every y-value, there is an x-value.
• Horizontal Line Test: Line parallel to x-axis intersects graph at least once.

Bijective Functions

• Both injective and surjective.
• Requires proofs for both injective and surjective to establish bijection.

Inverse Functions

• A function must be bijective to have an inverse.
• Notation: ( f^{-1}(x) )
• Properties:
  • Domain of ( f^{-1} ) is the range of f.
  • Range of ( f^{-1} ) is the domain of f.
  • Graph of ( f^{-1} ) is the reflection across the line ( y = x ).

Finding Inverses

• From a Mapping Diagram: Switch domain and codomain.
• From a Formula: Interchange x and y, solve for y.
• From a Graph: Reflect the graph across line ( y = x ).
• From Ordered Pairs: Swap x and y values.

Composite Functions

• A composite function is a function within another function.
• Notation: ( f(g(x)) ) or ( (f \circ g)(x) )
• Substitution: Substitute one function into another.
  • Example: ( f(g(x)) ) means substituting g(x) into f(x).
• Can compose a function with itself.