Lesson 25: Differentiating Between Relations and Functions

New Concept

A function is a mathematical relationship pairing each value in the domain with exactly one value in the range.

In a function, the dependent variable $y$ is a function of the independent variable $x$. In terms of the variables, $y$ is a function of $x$ and can be written like the following example:

What’s next

Next, you’ll master the basics: identifying functions with the vertical-line test, finding their domain and range, and writing your own function rules.

Property

The domain is the set of possible values for the independent variable (input values) of a set of ordered pairs. The range is the set of values for the dependent variable (output values) of a set of ordered pairs.

Examples

For the relation \{(2, 5), (3, 8), (2, 9), (4, 8)\}, the Domain is \{2, 3, 4\} and the Range is \{5, 8, 9\}.
For \{(-1, 'A'), (0, 'B'), (1, 'A'), (2, 'C')\}, the Domain is \{-1, 0, 1, 2\} and the Range is \{'A', 'B', 'C'\}.
If a plant grows 2 inches each week, the domain is weeks \((w)\) and the range is height \((h)\).

Explanation

Think of it like a vending machine! The domain is all the buttons you can press (inputs), and the range is all the possible snacks you can get (outputs). We only list each unique input and output once, no matter how many times they appear in a list. This keeps our sets tidy and accurate.

Property

A function is a mathematical relationship pairing each value in the domain with exactly one value in the range.

Examples

The set \{(1, 2), (2, 4), (3, 6)\} is a function because each input has only one output.
The set \{(1, 2), (1, 5), (2, 4)\} is not a function because the input 1 maps to both 2 and 5.
The equation $y = 3x + 1$ represents a function because any $x$ value you plug in gives you just one $y$ value.

Explanation

A function is a reliable rule where every input has exactly one, predictable output. Think of it like a loyal pet: when you call its name (the input), it comes to you (the output), and only you. A relation that pairs one input with multiple outputs is not a function—it’s just not that predictable or loyal!

Let's investigate if every input in this relation gets exactly one output—this is the crux of telling a function apart from just a general relation.

Example Problem:
Determine whether $\{(6, 5), (9, 4), (2, 2), (8, 7), (6, 1)\}$ represents a function.

Step-by-Step:

1. Examine the domain values: $6, 9, 2, 8, 6$. Notice that $6$ appears twice.
2. Check which range values the repeated domain value ($6$) maps to: $6 \rightarrow 5$ and $6 \rightarrow 1$.
3. Since $6$ maps to two different range values, the rule for functions is broken (each input should go to exactly one output).

Property

A graph on the coordinate plane represents a function if any vertical line intersects the graph in exactly one point.

Examples

A straight line like $y = x + 2$ passes the test, as a vertical line only ever hits it once.
A circle fails the test because a vertical line can slice through it at two points simultaneously.
A U-shaped parabola like $y = x^2$ passes the test because each vertical line crosses it only once.

Explanation

Imagine sliding a vertical ruler across a graph from left to right. If your ruler only ever touches the graphed line at a single point at any time, congratulations, you've got a function! But if the ruler hits the graph in two or more spots at once, it fails the test. It's a super quick, visual trick.