Functions, Domain, and Range
What is a Function?
A function is a special relationship between inputs and outputs where each input has exactly one output.
Think of a function like a machine: you put something in (the input), the machine processes it according to a rule, and you get something out (the output).
Formal Definition: A function $ f $ from set $ A $ to set $ B $ is a rule that assigns to each element $ x $ in set $ A $ exactly one element $ y $ in set $ B $. We write: $$ y = f(x) $$ where $ x $ is the independent variable (input) and $ y $ is the dependent variable (output).
The Key Rule: One Input → One Output
The most important thing to remember about functions is:
✅ IS a Function
Each input maps to exactly one output.
- $ f(1) = 3 $
- $ f(2) = 5 $
- $ f(3) = 7 $
Different inputs CAN have the same output — that's okay!
- $ f(2) = 4 $ and $ f(-2) = 4 $ ✅
❌ NOT a Function
One input maps to more than one output.
- $ f(1) = 3 $ AND $ f(1) = 7 $
The input 1 gives TWO different outputs — this violates the function rule!
If any single input produces multiple outputs, it is NOT a function.