Function concept; inverses; compositions - Mathematics: Analysis & Approaches IB Study Notes

Functions are mathematical relationships where each input (domain element) maps to exactly one output (range element). Formally, a function f from set A to set B is written as f: A → B, where A is the domain and B is the codomain. The range is the set of actual output values.

Function Notation: f(x) = y means input x produces output y. The independent variable is x; dependent variable is y.

One-to-One Functions (injective): Different inputs produce different outputs. No horizontal line intersects the graph more than once. If f(a) = f(b), then a = b.

Onto Functions (surjective): Every element in the codomain is an output. The range equals the codomain.

Inverse Functions: For f⁻¹ to exist, f must be one-to-one (bijective). The inverse reverses the mapping: f⁻¹(f(x)) = x and f(f⁻¹(x)) = x. To find f⁻¹: replace f(x) with y, swap x and y, then solve for y.

Composition of Functions: (f ∘ g)(x) = f(g(x)) means "apply g first, then f". The domain of f ∘ g is restricted by both functions' domains.

Key Formula: For reflection symmetry, f and f⁻¹ are symmetric about the line y = x.

Memory Aid: "DOmain is the DOor (inputs), Range is where you Ran (outputs reached), COdomain is where you COuld go (all possible outputs)."

Notation Standards: Always specify domain restrictions. Use proper function notation: f(x) not f.x or fx.

Think of functions as machines: you insert raw material (input) and receive a product (output). A coffee machine is a function: insert beans + water → output coffee. Each specific combination yields exactly one type of coffee.

Inverse Functions in Real Life: Temperature conversion is bijective. F = (9/5)C + 32 converts Celsius to Fahrenheit, while C = (5/9)(F - 32) is its inverse. Each Celsius temperature has exactly one Fahrenheit equivalent and vice versa. The functions "undo" each other.

Composition Applications: Currency exchange involves composition. Converting USD → EUR → GBP is (g ∘ f)(x) where f converts USD to EUR and g converts EUR to GBP. Order matters: converting USD → GBP directly differs from GBP → USD.

Encryption uses one-to-one functions. Your password (input) generates a unique encrypted code (output). The decryption function is the inverse, retrieving the original password. Without bijectivity, data recovery would be impossible.

Analogy for Composition: Think of nested boxes. f(g(x)) means x goes into box g, producing result a, which then goes into box f, producing final result b. You cannot open box f until box g has processed x.

Restricted Domains: f(x) = √x only accepts non-negative inputs (real numbers). Like a physical machine that accepts only certain-sized items, mathematical functions have domain restrictions to ensure real, defined outputs.

Practical Note: GPS coordinates use bijective functions—each location maps to unique latitude/longitude pairs, and each pair identifies exactly one location.