What’s a Function?

Other than being the most feared f-word in math teaching, a “function” is a question with only one answer.

Take the question: “How tall are you?”

We can change this to: “What’s the height of you?”

And if we wanted to compare your height to other people’s heights we can ask: “What’s the height of <insert person’s name here>?” This is the question template – the formula.

You can answer this question in inches, feet or cm, but the value of the answer remains unique, based on the person.

And that last little piece of the sentence is what makes the difference, based on the person.

The question changes with this little change. <cue music> This is the variable in the equation.

And we say, “Height is a function of the person.”

So where’s the fear come in?

As always, the notation is the kicker when it comes to teaching math.

Let’s change the question a little.

What’s the height of Enrique tomorrow if he grows three inches tonight?

Rather contrived, but work with me…

The question template is

What’s the height of <insert person’s name here> if he/she grows three inches tonight?

Which becomes

<height> = <height now> + 3


H = N + 3


And we haven’t even started with the f(x) stuff!

What’s this “domain” thing about?

Drawn in Autosketch + pasting in words from Excel
Image via Wikipedia

I wrote the first sentence of this post a little too hastily. A function has only one answer if there’s a valid question.

If you ask, “How tall is love?” someone will laugh at you. Or think you’re from California.

Our question template included some specifics that you don’t normally get:

What’s the height of <insert person’s name here>?

If we instead ask, “What’s the height of x?” we would then have to ask: what kinds of things can we put in for x? Can we put concepts, like love? Or just objects? The kinds of things that you can put in for x is called the domain.

For our question, we would need to specify that x is a person.

What do you think? How does this feel when explaining it to your kids?

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