Preview
IB DP Mathematics: Applications and Interpretation | SL & HL | Functions — Topic 2.7
📋 Learning Intentions
- Find the composite function \( (f \circ g)(x) \) by substituting one function into another and simplifying.
- Determine the domain and range of a composite function, including identifying restrictions that arise from the inner function's range and the outer function's domain.
- Find the inverse function \( f^{-1}(x) \) algebraically by swapping variables and solving, and verify using \( f(f^{-1}(x)) = x \).
- Apply domain restrictions to ensure a function is one-to-one before finding its inverse.
- Interpret the graphical relationship between a function and its inverse as a reflection in the line \( y = x \).
- Use the GDC to graph composite and inverse functions and verify algebraic results.
Real-World Introduction
A currency conversion app first converts euros to US dollars, then applies a transaction fee as a percentage of the dollar amount. Each operation is a function, but the output of the first becomes the input of the second — this is a composite function in everyday use.
Inverse functions appear whenever a process must be reversed. A pharmacist uses a dosage formula to find how much drug to administer given a patient's weight; the inverse of that formula recovers the patient's weight from a known dosage. Both directions of the relationship are mathematically meaningful.
🌍 Why This Matters In data science and machine learning, neural networks apply composite functions layer by layer to transform input data into predictions. The training process uses the inverse idea — backpropagation — to reverse the chain of operations and adjust parameters. Understanding composition and inversion is foundational to how modern AI systems learn.
Core Concepts & Definitions
When two functions are applied in sequence, the output of the first function becomes the input of the second. This chaining of operations produces a new, single function called a composite function.
📖 Key Definition — Composite Function Given two functions \( f \) and \( g \), the composite function \( f \circ g \) (read "f of g") is defined by: \[ (f \circ g)(x) = f(g(x)) \] The function \( g \) is applied first, then \( f \) is applied to the result. The composite \( f \circ g \) exists only for values of \( x \) in the domain of \( g \) for which \( g(x) \) lies in the domain of \( f \).
The order of composition matters: \( f \circ g \) and \( g \circ f \) are generally different functions. Think of \( f \circ g \) as "first do \( g \),