Determine and represent the domain and range of a function from a graph, table, set of ordered pairs, or context, distinguishing discrete from continuous and reasonable domains in real situations (A.F.2).

What this topic is asking

A.F.2 asks you to find and represent the domain and range of a function from a graph, table, set of ordered pairs, or context, and to tell discrete from continuous. On the Virginia Algebra I SOL these are Functions items: state a domain or range, pick the correct interval, or judge whether a situation is discrete or continuous. They appear as multiple choice, fill-in-the-blank, and hot-spot graph reads.

Domain and range defined

For a function, the domain is every value the input can take, and the range is every value the output actually reaches. Inputs are the $x$-coordinates; outputs are the $y$-coordinates. So for $\{(1, 4), (2, 7), (3, 4)\}$, the domain is $\{1, 2, 3\}$ and the range is $\{4, 7\}$ (the repeated output $4$ is listed once).

Reading from a graph

From a graph, scan the axes:

• Domain: how far the graph extends left to right along the $x$-axis.
• Range: how far it extends bottom to top along the $y$-axis.

A line that continues forever has domain "all real numbers" and range "all real numbers." A parabola $y = x^2$ has domain all real numbers but range $y \ge 0$ (it never goes below its vertex). Watch for endpoints: a closed dot includes that value, an open dot excludes it, the same convention as inequalities.

Discrete versus continuous

The distinction is about whether the input can take any value in an interval or only separate values:

• Discrete. The inputs are isolated, countable values, like the number of tickets sold ($0, 1, 2, \dots$) or people in a room. The graph is separate points.
• Continuous. The inputs fill an unbroken interval, like time, distance, or temperature, where fractional values make sense. The graph is a solid curve or line.

Representing domain and range

You can write a domain or range three common ways:

• Set notation (for discrete): $\{-2, 0, 1, 4\}$.
• Inequality notation: $x \ge 0$, or $-3 < x \le 5$.
• Interval notation: $[0, \infty)$ for $x \ge 0$, or $(-3, 5]$. A bracket includes the endpoint, a parenthesis excludes it, and infinity always takes a parenthesis.

Why context decides discrete versus continuous

Whether a domain is discrete or continuous is determined by what the input variable measures, not by the formula or the output. If the input counts things that come in whole units, people, tickets, cars, the input can only be $0, 1, 2, \dots$, so the domain is discrete and the graph is a scatter of points, even if the cost output is money with cents. If the input measures something that varies smoothly, time, distance, weight, then values like $2.7$ are legitimate, so the domain is continuous and the graph is an unbroken curve. This is why two situations with the same equation can have different domains: $C = 5n$ for the cost of $n$ shirts is discrete (you buy whole shirts), but $C = 5h$ for the cost of $h$ hours of parking may be continuous (you can park for $1.5$ hours). Reading the meaning of the input is the key skill, and it is exactly what the SOL's reasonable-domain items test.

How the SOL examines this topic

• Multiple choice. State a domain or range from ordered pairs or a graph, or judge discrete versus continuous.
• Fill-in-the-blank. Type a domain or range in interval or inequality notation.
• Hot spot / graph. Identify the extent of a graph along each axis.

Try this

Q1. State the range of $\{(0, 2), (1, 4), (2, 2)\}$. [1 point]

• Cue. Outputs: $\{2, 4\}$.

Q2. Is the number of students in a class a discrete or continuous quantity? [1 point]

• Cue. Discrete (whole numbers only).