"All limits, especially national ones, are contrary to the nature of mathematics… Mathematics knows no races… For mathematics the whole cultural world is a single country" – David Hilbert. "Face problems with a minimum of blind calculation, a maximum of seeing thought" – Hermann Minkowski

Domain and range (with two songs)

A Domain and Range song

See you graph a lot of functions but it breaks your heart
Never really had luck, couldn’t never figure out
the domain, and the range
But with this song that won’t last forever

On the next test you’re gonna put it together
the domain, and the range
The domain is all the x
Range is all the y. Let this song remind that sometimes
there are values that we can not plug in for x

When you have got a function that has an even root
a negative inside the radical won’t compute
So you can’t have a domain with any x-values
that give a negative under the radical

Another domain restriction that you gotta look for
Is zero can not be the denominator
Now you’re getting rid of any values for x
that make the bottom zero, the domain’s what is left

See you graph a lot of functions but it breaks your heart
Never really had luck, couldn’t never figure out
the domain, and the range
But with this song that won’t last forever

On the next test you’re gonna put it together
the domain, and the range
The domain is all the x
Range is all the y, Let this song remind that sometimes
there are values that we can not plug in, for x

And another

To the tune of “Irreplaceable” by Beyonce

To the right, to the right
To the right, to the right
Everything in the domain from the left to the right

On the graph, domain’s the x
From least to greatest, and here’s what’s next
I’m talking about the range — that’s the y
how far up and down the graph goes at the same time
Put those in order from least to greatest too
So you can answer the questions that I gave you

Sitting in Mr. D’s class, thinking bout how he’s such a fool,
How we’ll never ever forget how to do
domain and range yeah

You must now know domain, you must now know the range
I can help you understand in a minute
Matter fact, you’ll pass the test in a minute, baby

You must now know domain, you must now know the range
I can help you understand for tomorrow
So don’t you ever for a second ever forget, domain and range!

So when you have a coordinate point
A pair of (x, y), just remember this
Baby the first number’s the domain,
The second number’s the range
Cause the truth of the matter is… domain and range are so easy!