Domain and Range

What domain and range are

A function has two attributes that describe its scope. The domain is the set of all input values the function accepts. The range is the set of all output values the function produces over that domain.

Domain and range are attributes of a function.

Both belong to the function itself, not to the graph paper or the way someone happens to draw it. Change the function and the domain or range can change with it.

For a function with a real-world meaning, the domain is usually constrained by what the input can actually be. A school bus route's domain might be [0, 24] minutes: time can't run backwards, and the trip only lasts 24 minutes. A weekly temperature function's domain might be {Mon, Tue, …, Sun}, seven specific days, not the moments in between.

Discrete and continuous

The shape of the domain controls the shape of the graph.

A discrete-function graph is a set of separate points described by ordered pairs. The inputs are countable values (day-numbers, student counts, dice rolls); between two adjacent values, the function isn't defined. The graph is dots, not a line.

A continuous-function graph is a line that connects all points described by ordered pairs. The inputs flow without gaps (time, length, temperature, money in some contexts); between two nearby values, the function exists at every value in between.

The same five ordered pairs are shown two ways: as separated dots on the left, and connected by a line on the right. Which one is right depends on what the inputs represent. Day-numbers in a week → discrete (the function doesn't exist "halfway between Monday and Tuesday"). Time in minutes during a bus ride → continuous (every moment exists, and the bus is somewhere at each one).

See it: drag the input window

The widget below lets you select a portion of a function's domain and see the corresponding portion of the range. Drag the window endpoints along the x-axis (the input axis). The widget highlights the matching part of the graph and reads off the domain and range, both as numeric intervals and in plain words, since the curriculum requires both representations.

Use the bus-route preset first. The full domain is [0, 24] min and the full range is [0, 12] km. Slide the window narrower and the highlighted range shrinks too. Then switch to the daily-highs preset: the domain is now a SET of seven day-numbers, not an interval, and the range is a SET of seven temperature values. The widget changes its readout to match. Same widget, two different readouts for the two kinds of function.

Communicating in words

Numeric intervals like [0, 24] min are precise, but they're not the only way to describe a domain. The curriculum specifically asks students to communicate domain and range in words as well.

For the bus route:

• In intervals: domain [0, 24] min, range [0, 12] km.
• In words: "All times from 0 minutes up to 24 minutes" / "All distances from 0 km up to 12 km."

For the week of daily highs:

• In intervals (or sets): domain {1, 2, …, 7}, range {−8, −6, −3, −2, 1, 2, 5}.
• In words: "The seven day numbers (Mon = 1 through Sun = 7)" / "The set of recorded daily-high temperatures, ranging from −8 °C to 5 °C."

Both forms describe the same domain and range. The "in words" form is what you'd write in a sentence; the interval form is what fits on a graph axis.

Restrictions for real-world models

Functions in the abstract often have unrestricted domains. The formula y = 2x + 1 works for every real x. But functions that model something in the real world usually inherit restrictions from what the inputs can physically be.

• A school-bus route has domain [0, 24] min, not (−∞, ∞): time runs forward and the route ends.
• A puppy's weight as a function of weeks since birth has domain starting at week 0, not week −5.
• A water-tank fill function has domain ending when the tank is full.

The math doesn't care; the model does. When you describe a function's domain in words, mention the constraints that the real-world setting forces.

From a table to a graph

When the function comes as a table of values, plotting the points is a small but important step. Each row of the table is one ordered pair (x, y); each pair is one dot on the graph.

The next decision is whether to connect the dots — and this decision has a name, because getting it wrong is one of this unit's classic errors. The connect-the-dots reflex joins any plotted points with a line because "that's what graphs look like." Connect only if the function is continuous between the tabled values; leave them as dots if it's discrete.

A table of week-of-temperature readings stays as 7 dots: there's no temperature for "halfway between Monday and Tuesday." A table of bus-distance readings every 5 minutes can be connected, since the bus is somewhere at every instant between readings. Same table-to-graph process; the connect-or-not decision depends on what the inputs mean.

The plotter below ends on exactly that decision, and it doesn't decide for you. Plot the five rows, then make the call — and if you join the dots, watch what the line claims happened at 2.5 bookmarks.

Where it shows up in real life

A school-bus route time-vs-distance plot is the lesson's running example. Time runs from 0 to about 24 minutes (the trip's length); distance runs from 0 km at home up to about 12 km at the school. The domain reflects time's forward arrow and the trip's length; the range reflects how far the bus actually travels.

A freezer temperature log (say, the temperature inside a walk-in cooler at an Alberta grocery store, recorded every 15 minutes over a week) has a discrete domain (the time stamps the sensor wrote) and a continuous-looking range (a real-valued temperature reading). The plot is dots, even though the temperature itself was continuously varying. The graph is discrete because the measurements were taken at discrete times.

A prairie wheat-yield function of fertilizer applied per hectare has a domain restricted to non-negative rates (you can't apply negative fertilizer) and a practical upper bound where adding more stops increasing yield. Range covers the resulting bushels per hectare. The shape of both depends on real agronomic limits.

Worksheet

These aren't graded. Get them right, get them wrong. The goal is fluency with domain and range: in intervals, in words, and in the discrete-vs-continuous distinction.

Going further

Domain and range come back every time a function appears. In Grade 8 you'll formalize linear functions (y = mx + b); the default domain is all real numbers, but real-world restrictions often narrow it down. In Grade 9 you'll start drawing systems of two equations, where two functions intersect; the domain of the system is the overlap of the two individual domains.

The discrete-vs-continuous distinction you saw here returns later this strand under a different name. In the next Statistics lesson on Mean, Median, Mode, and Range, the same distinction applies to data: counts are discrete, measurements are continuous. The same idea, two different settings.