Units 1 & 2 (Functions and Relations)
Develop a deep understanding of functions as mathematical relationships, including their representation through equations, tables, and graphs. Study domain and range, function notation, and the behavior of different types of functions (linear, quadratic, exponential). Learn how to manipulate and transform functions, solve equations graphically and algebraically, and apply these concepts in modelling real-world situations.

Unit 1: Introduction to Functions and Relations

1.1 Understanding Relations and Functions

• Definitions: relation vs. function; one-to-one, many-to-one.

• Representations: sets of ordered pairs, tables, mapping diagrams, graphs.

• Domain & Range: from equations and graphs, including restrictions.

1.2 Types of Functions

• Linear Functions: gradient-intercept form, graphing from equation.

• Quadratic Functions: vertex form, symmetry, turning points.

• Exponential Functions: growth and decay contexts, asymptotes.

1.3 Function Notation and Evaluation

• Notation: f(x), g(x), composite expressions like f(2x), f(x + 3).

• Evaluation: substituting values, including nested and fractional inputs.

• Contextual Interpretation: real-world meaning of f(x) values and variables.

1.4 Graphing and Interpreting Functions

• Key Features: intercepts, intervals of increase/decrease, shape.

• Graphical Analysis: using calculator/technology to sketch and interpret.

• Real-World Contexts: modelling population growth, motion, cost.

Unit 2: Solving and Manipulating Functional Relationships

2.1 Solving Equations Involving Functions

• Linear and Quadratic Equations: algebraic methods and graphical checks.

• Exponential Equations: solving using logarithmic reasoning or technology.

• Intersection Points: solving f(x) = g(x) graphically and algebraically.

2.2 Transformations of Functions

• Vertical & Horizontal Shifts: f(x) ± a, f(x ± a).

• Reflections: in x-axis (−f(x)), y-axis (f(−x)).

• Stretch/Compression: af(x), f(ax) and their effect on shape and scale.

2.3 Composite and Inverse Functions (Introduction)

• Composition: (f∘g)(x), step-by-step substitution.

• Inverses: identifying if an inverse exists; finding inverse functions algebraically and graphically.

• Real-World Interpretation: reversible processes and input-output inversion.

2.4 Modelling and Problem Solving

• Worded Problems: translating into function form.

• Interpreting Solutions: domain restrictions, meaningful answers.

• Using Technology: to model, solve, and verify.