Preview
IB DP Mathematics: Applications and Interpretation | SL & HL | Topic 2 — Functions
📋 Learning Intentions
- Define a function and distinguish it from a general relation using the vertical line test and mapping diagrams.
- Use correct function notation including \( f(x) \), \( f: x \mapsto \), and composite notation \( f(g(x)) \).
- Determine the domain and range of a function from its equation, graph, or context, expressing answers in interval or set notation.
- Find the inverse function \( f^{-1}(x) \) algebraically and verify using the GDC.
- Interpret the graphical relationship between \( f(x) \) and \( f^{-1}(x) \) as a reflection in the line \( y = x \).
- Apply domain restrictions to ensure a function is one-to-one before finding its inverse.
Real-World Introduction
A vending machine accepts a coin and returns exactly one item. Press button B3 and you always get the same snack — never two different items, never nothing. This predictable, one-input-one-output behaviour is precisely what mathematicians formalise as a function .
Functions appear everywhere in applied mathematics: a currency converter maps one amount in euros to exactly one amount in dollars; a temperature model maps each hour of the day to exactly one temperature reading. The power of the function concept is that it guarantees predictability — given any input, the output is uniquely determined.
🌍 Why This Matters Every mathematical model in IB Math AI — from compound interest to logistic population growth — is built on functions. Understanding domain, range, and inverses allows you to determine where a model is valid, what outputs are possible, and how to reverse a calculation (e.g. finding the time needed to reach a target investment value).
Core Concepts & Definitions
Consider the mapping \( x \mapsto x^2 \). Every value of \( x \) produces exactly one value of \( x^2 \). Now consider the relation \( x^2 + y^2 = 9 \): for \( x = 0 \), both \( y = 3 \) and \( y = -3 \) are valid outputs. The second relation is not a function because one input produces two outputs.
📖 Key Definition — Function A function \( f \) is a rule that assigns to each element \( x \) in the domain exactly one element \( f(x) \) in the range . No input may produce more than one output.
Notation: \( f: A \to B \), where \( A \) is the domain and \( B \) contains the range. Equivalently written as \( y = f(x) \).
The key