Relations and Functions

Relations and functions are the bridge between chapter-01-sets and the broader mathematical landscape. While sets organize collections of objects, relations describe how objects from different sets connect to each other. Functions are special relations with a crucial constraint: each input maps to exactly one output. This chapter establishes fundamental concepts used throughout algebra, calculus, and applied mathematics. Understanding functions is essential because they model real-world phenomena—how temperature changes over time, how costs increase with quantity, or how populations grow.

Relations: Connecting Elements Between Sets

A relation describes a connection between elements from two sets. Think of it like a matching game. You have names in one set {Alice, Bob} and sports in another set {tennis, swimming}. A relation might connect Alice to tennis and Bob to swimming. We can also have Alice connected to both sports—that's allowed in a relation.

If we have set A = {1, 2, 3} and set B = {a, b}, a relation from A to B is any subset of ordered pairs (x, y) where x ∈ A and y ∈ B. The pair (1, a) means "1 is related to a."

Ordered pairs matter: (1, a) is different from (a, 1). Order signals direction and meaning.

Special Relations: Functions

A function is a special relation with a strict rule: every input has exactly one output.

Imagine a vending machine. You press button number 5 (input). Out comes exactly one item, say a chocolate bar (output). If button 5 sometimes gave chocolate and sometimes gave chips, it wouldn't be a function—it violates the "exactly one output" rule.

Formally, f: A → B is a function if for every element in A, there is exactly one corresponding element in B.

Domain: The set of all possible inputs (the first set).

Codomain: The set of all possible outputs (the second set).

Range: The set of outputs actually used. The range is a subset of or equal to the codomain.

Function Notation and Terminology

We write f(x) = y to mean "when you input x into function f, you get output y."

If f(x) = 2x + 3 and you input x = 5, then f(5) = 2(5) + 3 = 13. The function transforms 5 into 13.

Variable distinction: x is the independent variable (the input you choose), and y = f(x) is the dependent variable (it depends on what x is).

Types of Functions

One-to-One (Injective): Different inputs give different outputs. If f(a) = f(b), then a must equal b. Think of assigning unique ID numbers to students—each student gets exactly one ID, and each ID belongs to one student.

Onto (Surjective): Every element in the codomain is an output of some input. Nothing in the codomain is "unused." The range equals the codomain.

Bijective: Both one-to-one AND onto. A perfect pairing with no leftovers and no duplicates. These are the functions that have inverses.

Operations on Functions

Just as you add or multiply numbers, you can combine functions:

(f + g)(x) = f(x) + g(x): Add the outputs.

(f · g)(x) = f(x) · g(x): Multiply the outputs.

(f ∘ g)(x) = f(g(x)): Composition. First apply g, then apply f to that result. This is like a two-step process where one transformation follows another.

Inverse Functions

If a function is bijective, it has an inverse, written f⁻¹. The inverse "undoes" what the function does.

If f(x) = 2x + 3, the inverse is f⁻¹(x) = (x - 3)/2. Applying the function then its inverse returns you to where you started: f⁻¹(f(x)) = x.

To find an inverse: swap x and y, then solve for y.

Graphical Representation

The graph of a function plots (x, f(x)) as points on a coordinate system. The vertical line test determines if a relation is a function: if any vertical line crosses the graph more than once, it's not a function (because one input x would have multiple outputs).

Real-World Context

Relations model direct connections (like "person x supervises person y"). Functions model transformations and dependencies. Distance is a function of time, revenue is a function of sales quantity, student grades are functions of study hours. Functions let us describe patterns, make predictions, and understand cause-and-effect relationships in our world.

Key Formulas

• Function evaluation: y = f(x)
• Composition: (f ∘ g)(x) = f(g(x))
• Inverse condition: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x

Socratic Questions

1. In a function, why is the restriction that each input has exactly one output necessary? What would change in mathematics if we allowed one input to map to multiple outputs?

1. Consider the function f(x) = x². Is it one-to-one? Is it onto (if the codomain is all real numbers)? How would your answers change if the domain were restricted to non-negative numbers?

1. If you have two functions f and g, does f ∘ g always equal g ∘ f? Can you construct an example where they're different? What does this tell you about the order of operations?

1. Think about a real-world scenario: tracking student grades across multiple classes. How would you distinguish between a relation and a function? What information would you lose or gain with each model?

1. Why must a function be bijective for its inverse to exist? What goes wrong if you try to find the inverse of a function that's not one-to-one?