Functions

Summary and Exam Tips for Functions

Functions is a subtopic of Sequences, Functions, and Graphs, which falls under the subject Mathematics in the Edexcel IGCSE curriculum. Functions are mathematical constructs that map inputs to outputs, often represented using function notation such as $y = f(x)$ or $f: X \to Y$. Here, $f$ denotes the function, $x$ is an element from the domain $X$, and $y$ or $f(x)$ is an element from the range $Y$. Understanding domain and range is crucial, as they represent all possible input and output values, respectively. Functions can be one-to-one or many-to-one mappings, which can be visualized using mapping diagrams.

Composite functions involve combining two functions, denoted as $f(g(x))$, where the order of operations is significant. Inverse functions reverse the effect of the original function, existing only if the function is one-to-one. To find an inverse, swap the roles of $x$ and $y$ and solve for the new $y$. Practice problems often involve determining domains, ranges, and inverses, as well as evaluating composite functions.

Exam Tips

• Understand Function Notation: Familiarize yourself with how functions are represented and practice converting complex expressions into function notation.
• Domain and Range: Always determine the domain and range for each function, especially when dealing with composite and inverse functions.
• Composite Functions: Pay attention to the order of operations in composite functions. Remember that $f(g(x))$ means you apply $g$ first, then $f$.
• Inverse Functions: Practice finding inverses by swapping $x$ and $y$ and solving for the new $y$. Ensure the function is one-to-one before finding its inverse.
• Practice with Diagrams: Use mapping diagrams to visualize function relationships and understand one-to-one and many-to-one mappings effectively.

By mastering these concepts, you'll be well-prepared for questions on functions in your exams.