Functions, Domain, and Range

Understand function notation and find domain and range. Math 10-C Alberta mathematics curriculum.

Lesson 7.2 of Relations and Functions in Math 10C — Alberta curriculum lessons.

What you'll learn in this lesson

• What Makes a Relation a Function?. Learn the one strict rule that separates functions from general relations, and test it with arrow diagrams.
• Testing Functions in Points, Diagrams, Tables, and Words. Apply the one-input-one-output rule to ordered pairs, tables, and real-world word problems.
• Domain and Range. List the domain and range of a function from a set of points or from a coordinate graph.
• Discrete vs. Continuous and the Vertical Line Test. Distinguish discrete from continuous data and use the Vertical Line Test to identify functions from graphs.

Inside the lesson — a free preview

The Definitions

The One Rule — Each Input Has Exactly One Output. The golden rule of functions: each element of the domain must connect to only one element of the range.

One input → one output. Not every relation is a function!

An input is allowed to share an output with other inputs. What is not allowed is one input pointing to two different outputs.

Analogy — The Vending Machine. Think of a vending machine. You press button B4 and you always get the same snack. That's a function.

If pressing B4 sometimes gave chips and sometimes gave a chocolate bar, it would not be a function — one input, two possible outputs.

Test with Ordered Pairs

The Rule for Ordered Pairs. A relation is a function if no x-value repeats with a different y-value.

Scan the first numbers (inputs). If every first number is unique, it is a function. If the same first number appears twice but with different second numbers, it is not a function.

Example A — Function. Relation: \(1,6),\ (2,4),\ (3,5),\ (4,6),\ (5,5)\

Check the x-values: 1,\ 2,\ 3,\ 4,\ 5 — all different.

Notice 6 and 5 both appear as outputs — that is fine, repeated outputs are allowed.

This is a function.

Example B — NOT a Function. Relation: \(4,2),\ (6,2),\ (6,3),\ (8,2),\ (9,3)\

Check the x-values: 4,\ 6,\ 6,\ 8,\ 9 — the input 6 appears twice.

Input 6 gives output 2 once and output 3 once — one input, two different outputs.

This is NOT a function.

Example C — Function (bonus pattern). Relation:

\(1,3),\ (2,5),\ (3,7),\ (4,9),\ (5,11),

(6,13),\ (7,15),\ (8,17),\ (9,19),\ (10,21)\

All x-values 1 through 10 are different.

This is a function. These pairs follow the rule y = 2x + 1.

Worked example: Worked Example: Two Arrow Diagrams

Look at these two relations connecting numbers to vehicles.

Relation A — Is a Function. Each input (1, 2, 3, 4) has exactly one arrow leaving it.

This is a function.

Relation B — Also a Function. Each input (the vehicles) has exactly one arrow leaving it — even though Bicycle and Car both point to 2.

Two different inputs sharing one output is allowed.

This is also a function.

Key Insight. Two different inputs are allowed to give the same output.

What breaks a function is one input giving two outputs.

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