The Language of Functions
Mastery of VCE Mathematical Methods begins with a mastery of its language. This section establishes the non-negotiable vocabulary for relations, functions, domain, and range.
Relations vs. Functions
A relation is any set of ordered pairs (x, y). A function is a special type of relation where each input x maps to at most one output y.
The Vertical Line Test:
If a vertical line intersects a graph more than once, it is not a function.
Domain and Range
Domain (D): The set of all permissible input (x) values.
Range (R): The set of all resulting output (y) values.
Formal Function Notation
A function f is formally defined using the notation f: D \to \mathbb{R}, f(x) = \text{rule}. This precisely communicates its name, domain, co-domain, and rule.
Core Function Families
Explore the characteristics of core function families. Select a function type to see its graph and key features, and use the sliders to see how parameters affect its shape.
Select a Family
Key Features
Details will appear here.
Interactive Transformations
Manipulate the parameters a, n, h, and k in the equation y = a \cdot f(n(x-h)) + k to see how each one affects the graph of a parent function in real-time. Remember: Dilations & Reflections first, then Translations.
Controls
Transformed Equation
Combining & Inverting Functions
Functions can be used as building blocks. This section explores composite functions and the conditions required for a function to have an inverse.
Composite Functions: f(g(x))
A composite function f(g(x)) is formed when the output of an inner function, g(x), becomes the input for an outer function, f(x).
Condition for Existence: ran(g) \subseteq dom(f)
Inverse Functions: f^{-1}(x)
An inverse function f^{-1}(x) reverses the action of the original function f(x). An inverse exists only if the original function is one-to-one (passes the Horizontal Line Test).
Key Property: dom(f^{-1}) = ran(f)
Modelling & Exam Strategy
The ultimate goal is to apply this knowledge to solve problems. This section summarises key strategies for assessment tasks and examinations.
The Mathematical Modelling Process
- Understand the Context: Deconstruct the problem to identify key variables and relationships.
- Choose a Function: Select an appropriate function family to model the relationship.
- Formulate the Model: Determine the specific equation, parameters, and constraints (the domain).
- Solve and Interpret: Use the model (and calculus) to answer questions and interpret the results in context.
Exam 1: Tech-Free
- Focus on speed and accuracy in algebra.
- Practice core skills (differentiation, factorisation).
- Set out working logically and clearly.
Exam 2: Tech-Active
- Become an expert with your CAS calculator.
- Focus on interpreting the question and setting up the model.
- Prepare a high-quality, well-organised bound reference.
