Domain and Range, and the Inputs a Situation Allows

In this lesson you will: State the domain and range of a function given as a set, a table or a graph; Cut the domain down to the inputs a real situation can actually supply; Explain why an input outside the domain produces an answer that is not viable.

Two sets, and which axis each one lives on

The domain is the set of inputs the function accepts. The range is the set of outputs it produces. For a set of ordered pairs both are easy: collect the first coordinates for the domain, the second coordinates for the range, and list each value once.

On a graph, read the domain by sweeping your eye left to right along the horizontal axis — how far does the picture extend? Read the range by sweeping bottom to top. Endpoints that are marked with a solid dot are included.

The situation gets a vote

A rule written on its own will usually accept any number you feed it. C(b)=310b happily returns a value for b=-4 and for b=2.9. The situation those symbols describe may refuse both.

If b counts buses, then b has to be a whole number and it cannot be negative. Those are not algebraic restrictions; they come from the story. This is the same judgment you made in Chapter 5 when you asked whether a solution to an inequality was viable — here you are making it before you compute, by choosing the domain.

Where the domain stops, the range stops

Changing the domain changes the range, because the range is only ever the outputs the allowed inputs produce. Concert tickets at $14 each, with a limit of six per order, give T(n)=14n with domain \0,1,2,3,4,5,6\.

The range is then \0,14,28,42,56,70,84\ — seven values, not an interval. The rule 14n would produce every multiple of 14 forever; the order limit stops it at 84.

Common mistake — Giving the domain of the formula when the question asked about the situation

Asked for the domain of C(b)=310b, the cost of renting b buses, a student answers \"all real numbers.\"

For the bare formula that answer is correct, and it is the answer every textbook gives for a rule with no story attached. That is exactly why it gets written down without a second thought.

But the question named buses. A school cannot rent -2 buses, and it cannot rent 2.9 of one. The domain in this situation is the whole numbers 0,1,2,3,..., possibly with an upper limit if only so many buses exist.

The reverse error also happens: restricting a domain that has no reason to be restricted. If x is a temperature change, negatives are entirely ordinary and excluding them loses half the answers.

How to catch it: Take the most extreme value your claimed domain allows — the most negative, the least whole — and put it into the sentence, not the formula. \"The school rents -2 buses\" fails out loud. If the sentence is nonsense, that value is not in the domain.