Functions

A large portion of what you’re doing in the early sections of this mathematics curriculum is learning to read. Keep that in mind. When you see the word “dog” many things immediately come to mind: quadrupedal mammal, friendly, loyal, perhaps your own experiences with a dog. Similarly, when I see $f(x) = \sqrt(1-x^2)$ I immediately see a half circle with radius centered on the origin above the x-axis.

Our goal is that you develop a level of fluency of reading elementary functions such that you immediately see these things as well. This will require two things from you: 1. you must learn your basic function classes and 2. you must learn transformations on your basic functions.

Often, we do not fully express the notation for a function because it would be onerous. We suppress notation when the sets we are working with are clearly understood.

We often write a function as $f(x) = y$. It is not uncommon in higher mathematics to write $f : \text{Domain} \rightarrow \text{Range}$. Here, x and y are serving as place holders. x usually specifies any element in the domain and y is often replaced with an explicit rule. Generally, we have total control of the domain and while we will see we may need to restrict the domain for various considerations that we will shortly become acquainted with, it is easy to specify a reasonable domain. On the other hand, even if we have a range in mind, it might not be obvious what the actual range is without careful study of our functional rule!

Let us consider $f : \mathbb{N} \rightarrow \mathbb{N} \ni : f(x) = 2x$. Recall that $\mathbb{N}$ is shorthand for the set of natural numbers. What does $f$ do here? It is pretty clear that this function works for all natural numbers, but notice it does not return all the natural numbers. It is true that if I start with a natural number and multiply it by two that I again obtain a natural number. Here, I might define $\mathbb{E}^+$ as the set of even, positive numbers. Then $f : \mathbb{N} \rightarrow \mathbb{E}^+$.

Let’s consider an alternative definition of a function. A function takes an element of the domain and assigns it exactly one element of the range. The second part of this sentence is crucial. We put in an input and we receive only one output. This is useful because functions describe the observed world we live in. Suppose you are at the grocery store and you have two identical packs of gum. One is scanned and rings up as \$1.00 and the next is scanned. What do you expect it to ring up as?

Now, suppose you scan it and it rings up as \$27.00. There must be something wrong! You scan it again and it rings up as \$310.11. You keep scanning it and it just rings up at different prices. Then you rescan the original one and the same thing happens except this time it reads out a long list of prices the next you time you scan it. Which one do you pay? This would be what would happen if the grocery store scanner was not using a function! Moreover, this function takes the domain: the set of products at the store to the range: the set of prices for those products. Notice that the sets can be totally different.

Does the above chart represent a function?

Yes, elements in the range may be repeated. It’s fine that two different items have the same price.

Does the above cartesian graph of a circle represent a function?

No. For most x-coordinates on the cartesian plane, we see two y-coordinates.

Does the above chart represent a function?

No. If we regard the symptoms as our domain, any individual symptom may indicate many diseases.

Does the above graph represent a function?

Yes, this half circle is a function and it teaches us something valuable. For most collections of points, if we restrict our output set we can often create a local function.

The idea of a linear relationship between two variables is so fundamental we will return to it again and again in mathematics and in all your subsequent science classes. Linear relationships are almost biologically hardwired into the human brain. People generally have a good intuition for these sorts of relationships and often hope that intuition carries into other areas of mathematics. This is neither good nor bad, but we should use it as a jumping off point to build what I will call a mathematical intuition. This is very different from your standard intuition. We will start to see good examples of this as learn to look at algebraic expressions and see geometric curves.

$ax + by + c = 0$

is the general form of the equation of a line. It is also, not terribly helpful. If manipulate this function algebraically, we obtain

$ax + by = c$

$by = -ax – c$

$y = \frac{-a}{b}x – \frac{c}{b}$

which can be rewritten in the following more famous form:

$y = mx + B$

Here, m is the slope of the line and (0, B) is the y-intercept.