The Atomic Functions
You might get the impression from what we've done so far that there's more functions than you could ever hope to learn. To some extent, you're right. There are infinitely many functions. However, the most commonly used functions are just combinations of a handful of functions I like to call atomic functions. These are the functions that Geogebra and other mathematics software most commonly use to model the natural world. We're going to spend the rest of this chapter getting to know a little bit about the algebra involved with working with these atomic functions.
If you really don't want to learn any algebra, you can skip the remaining sections of this chapter and jump ahead to Limits. The penalty is that if you don't understand how the atomic functions work together to build more complex functions, you won't be able to learn the algebraic calculations of derivatives and integrals. You can absolutely learn the concepts of calculus without doing so, but obviously you won't learn the algebra. The choice is up to you.
OK...for those of you still with me, the Geogebra sheet below shows all the atomic functions. Click the dot next to each to turn them on and off. Each function has an informal name. For example, the function that is visible at first is called the "identity" function. Click the dots on and off to see the others.
Don't worry if you don't know much about e, sine or cosine. We'll talk more about them when we need to later.
Just a few things to comment:
- Constant functions can be any constant. It doesn't have to be 3 like above. It could also be -5.64, or 72 or . Try it by typing h(x)=4 into the input bar, creating a function that is flatlined at y=4.
- There are power functions for any power. I've shown you the power functions with powers 2, 3, and 4, but any number can be used for a power. For instance is a power function.
Quick Check: Consider the function f(x)=19. What's f(18)?
In the next lesson we'll see how to combine the atomic functions to build more complex functions.