Learn what relations and functions are, how to test for a function using ordered pairs and the vertical line test, and why every function is a relation but not every relation is a function.
Lesson 1.1 of Quadratic Functions & Equations in Math 20-1 — Alberta curriculum lessons.
Definition: Relation. A relation is any set of ordered pairs. It is simply a collection of (x, y) points that are connected in some way.
Relations can be represented in four different ways:
Definition: Function. A function is a special kind of relation where no two ordered pairs share the same x-coordinate. For every x-value, there is exactly one y-value.
Every x-input goes to exactly one y-output. You cannot have one x pointing to two different y values.
Look at the x-values in the set. If any x-value repeats with a different y, the relation is NOT a function.
Example A — Not a Function. \(1,2),\ (1,3),\ (2,4),\ (3,5)\
Example B — Function. \(1,2),\ (2,3),\ (3,4),\ (4,5)\
The Vertical Line Test.
Key Rule to Remember. Every function is a relation, but not every relation is a function.
Domain and Range.
Memory Trick. D comes before R in the alphabet, just like x comes before y.
There are two standard ways to express domain and range.
Set-Builder Notation.
Interval Notation.
Definition: Function Notation. Instead of writing y in an equation, we use function notation to name the function and make the input variable explicit.
Why Use Function Notation?.
Example 2 — Write in Function Notation. Write y = (1)/(2)w^2 + 3w + 5 in function notation.
For each graph below, identify the domain and range using set-builder notation.
Solution (a) — Diagonal Line.
Solution (b) — Square Root Curve.
Solution (c) — Horizontal Line.
Solution (d) — Shifted Square Root Curve.
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