Study Notes
Functions are mathematical rules that transform inputs into outputs, often represented using function notation. They can be simple or involve operations like inverses and composites.
- Function Notation — a symbolic representation of a function. Example: f(x) = x² means the function f maps x to x squared.
- Mapping Diagram — a visual representation showing how each element of the domain is paired with an element in the range. Example: A one-to-one mapping diagram pairs each input with a unique output.
- Domain and Range — the set of all possible inputs (domain) and outputs (range) of a function. Example: For R = {(1, 2), (2, 2), (3, 3), (4, 3)}, Domain = {1, 2, 3, 4}, Range = {2, 3}.
- Composite Function — a function made by combining two functions. Example: If f(x) = x² and g(x) = x + 2, then f(g(x)) = (x + 2)².
- Inverse Function — a function that reverses the effect of the original function. Example: If f(x) = y, then f⁻¹(y) = x.
Exam Tips
Key Definitions to Remember
- Function Notation: f(x) represents a function of x
- Domain: Set of all possible input values
- Range: Set of all possible output values
- Composite Function: Combination of two functions
- Inverse Function: Reverses the original function
Common Confusions
- Mixing up domain and range
- Incorrect order in composite functions
- Assuming all functions have inverses
Typical Exam Questions
- What is the domain of f(x) = x²? Domain is all real numbers
- Find the inverse of f(x) = 2x + 3? f⁻¹(x) = (x - 3)/2
- What is f(g(x)) if f(x) = x² and g(x) = x + 1? f(g(x)) = (x + 1)²
What Examiners Usually Test
- Understanding and using function notation
- Identifying domain and range
- Forming and evaluating composite functions
- Finding and verifying inverse functions