Study Notes
A function is a special type of relation where each element in the domain is associated with exactly one element in the range.
- Cartesian Product — the set of all ordered pairs from two sets. Example: If A = {1, 2} and B = {3, 4}, then A×B = {(1,3), (1,4), (2,3), (2,4)}.
- Relation — a subset of a Cartesian product. Example: If A = {1, 2} and B = {3, 4}, then R = {(1,3), (2,4)} is a relation from A to B.
- Function — a relation where each input has exactly one output. Example: F = {(1,5), (2,4), (3,5)} is a function from A = {1, 2, 3} to B = {4, 5, 6}.
- Domain — the set of all possible inputs for a function. Example: For f(x) = sin(x), the domain is all real numbers.
- Range — the set of all possible outputs of a function. Example: For f(x) = sin(x), the range is [-1, 1].
- Composite Function — a function made by combining two functions. Example: If g(x) = 2x + 3 and f(x) = x^2 - 3x + 1, then (g◦f)(x) = 2(x^2 - 3x + 1) + 3.
- One to One Function — each element of the domain maps to a unique element of the range. Example: f(x) = 2x is one to one.
- Onto Function — every element of the range is mapped by some element of the domain. Example: f(x) = x^2 is onto if the range is non-negative real numbers.
- Inverse Function — reverses the effect of the original function. Example: If f(x) = 4x - 8, then f^(-1)(x) = (x + 8)/4.
Exam Tips
Key Definitions to Remember
- A function is a relation where each input has exactly one output.
- The domain is the set of all possible inputs.
- The range is the set of all possible outputs.
- A composite function is a combination of two functions.
- A one to one function maps each input to a unique output.
- An onto function covers the entire range.
- An inverse function reverses another function.
Common Confusions
- Confusing the domain with the range.
- Assuming a function can map one input to multiple outputs.
- Mixing up one to one and onto functions.
Typical Exam Questions
- What is the domain of f(x) = 1/x? The domain is all real numbers except x = 0.
- Is the function f(x) = x^2 one to one? No, because different inputs can produce the same output.
- Find the inverse of f(x) = 3x + 2. The inverse is f^(-1)(x) = (x - 2)/3.
What Examiners Usually Test
- Understanding of function definitions and properties.
- Ability to find domains and ranges.
- Skill in composing and inverting functions.