What Are Domain and Range?

Before we dive into quadratic functions specifically, let's make sure we understand two foundational concepts that apply to all functions and relations.

Domain

The domain of a function is the set of all permissible values that $x$ (the input) may take.

Range

The range of a function is the set of all permissible values that $y$ (the output) may take.

Think of it this way:

Domain = all the $x$ -values the function can accept (the horizontal extent of the graph)
Range = all the $y$ -values the function can produce (the vertical extent of the graph)

Analogy

Imagine a vending machine. The domain is like all the buttons you're allowed to press (your inputs). The range is all the possible drinks or snacks that can come out (the outputs). You can't get an output that the machine doesn't contain, and you can't press a button that doesn't exist!

Interval Notation Refresher

We describe domain and range using set notation or interval notation. Here's a quick guide:

$x : x \in R$ means " $x$ can be any real number" — this is the most common domain for quadratics
$y : y \geq 3$ means " $y$ can be 3 or any value greater than 3"
$y : y \leq 5$ means " $y$ can be 5 or any value less than 5"

You may also see interval notation:

$(- \infty, \infty)$ means all real numbers
$[3, \infty)$ means $y \geq 3$ (square bracket means the endpoint is included)
$(- \infty, 5]$ means $y \leq 5$

Exam Tip

Square bracket $[;]$ means the value is included (like $\geq$ or $\leq$ ). Round bracket $(;)$ means the value is not included (like $>$ or $<$ ). We always use round brackets with $\infty$ because infinity is not an actual number we can reach.

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