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By the end of these notes, you will be able to:
Before we talk about functions, we need to understand mappings.
A mapping is simply a rule that connects input values to output values. Think of it like a machine: you put a number in, the rule does something to it, and a number comes out.
Example: The rule x↦x+1 (read as "x is mapped to x + 1") takes any number and adds 1 to it.
| Input (x) | Output (x+1) |
|---|---|
| 1 | 2 |
| 2 | 3 |
| 3 | 4 |
| 4 | 5 |
This can also be drawn as a mapping diagram — two ovals connected by arrows. The left oval holds the input values, and the right oval holds the output values. Arrows show which input connects to which output.
There are three types of mappings. Understanding them is essential because only some of them are functions.
Definition: Each input value maps to exactly one output value, AND each output value comes from exactly one input value.
Think of it as a perfect pair — every input has its own unique output, and no two inputs share the same output.
Example: x↦x+1
Every input gives a different output. This is one–one.
Graphically: If you draw a horizontal line anywhere across the graph, it will cross the curve at only one point.
Definition: Two or more different input values map to the same output value.
Example: x↦x2
Two different inputs give the same output. This is many–one.
Graphically: A horizontal line crosses the graph at more than one point (for at least one horizontal line you draw).
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