What’s the Domain, Why You Need It and How You Get It

As you teach domain and range, do you get the question, “Why are we doing this?”

No doubt the question, “When am I ever going to use this?” comes up too, right?

I’ve asked myself that question my whole teaching and tutoring life. And now’s the time for an answer.

A function is really a question.

As I wrote in a previous post, a function is a question with only one answer to a valid question.

When I write:

y = 3x + 2 where x = 4

I mean: “What is three times a number (that number is four), plus two?”

The domain is all the possible questions:

What is three times a number (that number is five,) plus two?
What is three times a number (that number is six,) plus two?
What is three times a number (that number is seven,) plus two?
What is three times a number (that number is eight,) plus two?
<how long will I have to do this – Egad!>

Not only do the questions go on forever, but they also have all the fractions and decimals in between. And all the negative versions of those numbers (and zero).

So, what’s the point, you might ask. Looks like the questions go on forever and you can just pick any number.

The domain might not include all the numbers.

The two sticky points for the definition of “function” are bolded:

A function is a question with only one answer to a valid question.

The “valid question” part is where the domain comes in.

The numbers that make “valid” – meaning we actually can get some answer – are the numbers that aren’t negative.

Many functions have “all real numbers” as a domain. There are no limits on the things you can put in, other than numbers that aren’t imaginary or alligators.

For the most part, there are only two places where you have to be careful of limited domains. Those are

Numbers that cause a zero to turn up in the denominator
Numbers that cause negatives to turn up in square roots.

Here are two videos tackling each:

What do you think? Is this easier to teach when you consider “analyzing” the function rather than “solving” it? Share your thoughts and tips in the comments!

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