Description of the Topic: The topic Functions and Relationships focuses on the definition, interpretation, and representation of different types of functions and their graphs. Learners will study how to analyse, transform, and compare functions, as well as solve equations and inequalities derived from these functions. Emphasis is placed on identifying key features, domain and range, and the relationship between algebraic expressions and graphical representations.

Subtopics:

1. Definition and Representation of Functions
2. Types of Functions and Graphs
3. Transformations of Functions
4. Inverses of Functions
5. Solving Equations and Inequalities

Learning Objectives

1. Definition and Representation of Functions

• Objective: Learners must understand the concept of a function and be able to represent it in different forms.
• Sub-objectives:
  • Define a function in terms of input and output values.
  • Represent functions using equations, tables, and graphs.
  • Distinguish between relations that are functions and those that are not.
  • Identify the domain and range of a function.

2. Types of Functions and Graphs

• Objective: Learners must be able to identify and work with different types of functions covered in the curriculum.
• Sub-objectives:
  • Recognise and sketch graphs of linear, quadratic, exponential, hyperbolic, and trigonometric functions (as applicable for Grade 10–12).
  • Compare characteristics of different types of functions.
  • Identify intercepts, turning points, asymptotes, and axes of symmetry where applicable.

3. Transformations of Functions

• Objective: Learners must understand and apply transformations of basic functions.
• Sub-objectives:
  • Translate functions vertically and horizontally.
  • Reflect functions in the x-axis or y-axis.
  • Stretch and compress functions vertically or horizontally.
  • Describe and represent the effect of transformations on graphs.

4. Inverses of Functions

• Objective: Learners must be able to determine and interpret the inverse of a function.
• Sub-objectives:
  • Define an inverse function and its relationship to the original function.
  • Determine the inverse of a given function algebraically and graphically.
  • Understand conditions for a function to have an inverse (one-to-one functions).
  • Sketch the graph of an inverse function.

5. Solving Equations and Inequalities

• Objective: Learners must solve equations and inequalities using knowledge of functions and their properties.
• Sub-objectives:
• • Solve linear, quadratic, and exponential equations.
  • Solve inequalities graphically and algebraically.
  • Interpret solutions of equations and inequalities in the context of functions.
  • Apply graphical methods to identify the points of intersection of functions.