05.09.2019: Function Notation and Operations, 1/4: Function Notation

Today's soundtrack is Insomnium: Above the Weeping World, a melodic death metal album.

Today, I'm staring the final unit of the final module of this course. Hooray, hooray! This lesson is about function notation (not to be confused with factorial notation).

We begin by reviewing relations and functions. Relations are sets of numbers that go together. They can be represented with a table, a graph, or a listing of R={...}. The domain of a relationship is its x-values (its first values in any ordered pairs); its range is its y-values (the second values of any ordered pairs). Any duplicate values need not be recorded more than once.

Next, functions. A function is a kind of relation. Like all other relations, it has a domain and a range; it can be represented in many ways, etc etc. BUT - a function will only have one output for any input. Any input (x-value) will only generate one output (y-value). If we look at the graph of a function, we can draw a straight vertical line through any point on the graph and we will never see more than one intercept on that graph. We call this the vertical line test.

As I mentioned, there are several ways to represent a function - graphs, tables, etc. We can represent a function in an equation like this: y=2x+1. We could also say f(x)=2x+1, since y is the function of x's value. And if we wanted to find out what y equals when x = 3, we could say "Find the value of y if x has the value of 3, using the equation f(x)=2x+1." But there's another way to do this: a more compact, efficient way: function notation. In function notation, we put the value of x that we want to solve for inside of the brackets on the left side of the equation. So instead of saying "Find the value of y if x has the value of 3, using the equation f(x)=2x+1," we could just say "Solve f(3)=2x+1." Then we would substitute 3 for all instances of x in the equation, simplify, and hand in the paper, job done, time to grab a root beer and play some Rocket League. Wouldn't that be nice? But no, this lesson isn't done yet. On we go.

We may be asked to find f(x)=_, which is asking us the opposite question from that posed above. Instead of asking us "what is y when x = _," we are being asked to determine what the value of x is when y = the given value. To solve this kind of equation, we simply switch back out of function notation, substitute the given value for x, and solve algebraically, then hand in your paper, grab a Dr. Pepper, and play some World of Warcraft.

But how do you kill that which has no life?