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Mathematical Journeys: The Function Machine

y = f(x)

I can't tell you how many times I've had students come to me profoundly confused about their entire math unit, all because their teachers never fully explained this concept. Teachers throw this equation up on the board without discussion as if it explains everything – which it does, but only if you know what it means. So let's discuss!

First off, it's important to remember that this is not just an equation; it's an indication of a larger concept. We'll get to that in a minute, but let's start at the beginning.

Imagine that I have a little machine which I set on the table in front of you and turn on. You place a number in the slot in the top, and the machine begins to hum and churn. After a few moments, a drawer opens at the bottom and you pull out a different number. You can repeat this with any number you like, any number of times.

Now this is a single-purpose machine, which means it has one rule that it uses to transform the starting number into the final number. When you insert a number, it applies the rule to it, and presents the result. For example, say my machine's rule is that it takes the input number and adds 4 to it. If you give it a six, it will give you a 10. If you give it a 3, it will give you a 7. No matter what number you give it, it will add four and then give you the result. In math terminology, we call machines like this “functions.”

Definition: A “function” is a machine that transforms one value into another value via a set rule.

The notation for a function is

f(x)

where x is the input value. When you see an equation in the form

f(x) = x + 4

That is simply defining the rule of the function. In this case, this is the mathematical notation for our x + 4 machine from earlier. A common question type when learning about functions is as follows:

f(x) = 3x + 6. What is f(5)?

Don't panic! This question is simply asking you to send a number of their choosing through your machine. In this case, the questioner has handed you a five and a machine with the rule 3x + 6, and wants you to give him his result. You just substitute 5 for the x in the rule and solve it. In our case,

3(5) + 6
15 + 6
21

So the answer is 21.

An important thing to note at this point is that the “f” in f(x) is NOT a variable like x or y. “f” is an OPERATOR, like a plus sign or a division bar. It's an indication of an operation being performed. You could replace the “f” with literally anything, so long as you define what the rule of the machine is so you know it's a function. The full notation for this operator is actually “f()”, and you put your starting value inside the parentheses. You'll see this a lot on the SAT, where they include questions which replace the “f” with a series of nonsense symbols in order to test your understanding of the basic concept. They may even leave out the parentheses to confuse you further. On the SAT, you might see a question like this:

@ is a function such that @x = 2x – 3. What is @6?

Don't be thrown by the use of an @ sign; they're just trying to confuse you. Read the rest of the question, and you'll see that they specifically define it as a new operation, a function with a rule they've given you. It's the same kind of question we just did; they want you to plug the 6 in to the rule. Heck, the operator could be a little picture of an elephant for all you care; what's important is that they define the rule of the function so that you know what effect it has on the input value.

Now, what about that pesky f(x) = y ?

Okay, bear with me, we're going on a journey here. You have your machine, and you know that each input number you put into it will present you with a corresponding result number. We've been able to pick out individual pairs of corresponding values, but what if we wanted to represent the entirety of this transformation as a graph, something that would show all possible pairs at once, even the nitpicky ones in between integers?

NOTE: For the time being, we're going to restrict our discussion to only rules of the first power (that means no exponents). Things get a bit more complicated if you've got powers in your rule, so we'll save that for later.

If your rule is a first-power rule, then for every input value there's going to be exactly one output value. Let's take our original x + 4 machine again. Say I put in a number 2, and it gives me a 6. Now, if I take the 2 and the 6 out of the machine, I'll have a pair of corresponding numbers that I can plot as a point on a two-dimensional graph (one axis for the starting values, and the other for the results). We've been calling our starting value x in the equation, so let's make the starting value our x coordinate. That would make our resulting value the y coordinate. So our example above gives us the ordered pair (2, 6). Plug in some more x values, and you'll get more y values and be able to plot more points, and the shape of the graph will begin to appear. In other words, the result of the transformation is the y value for any given value of x.

And THAT can also be written as f(x) = y.

So f(x) = y is not actually an equation as much as an indication of a concept. Basically, we're saying that we can replace the phrase “f(x)” with the variable “y”, because we know that the number we'd get would be the corresponding y coordinate for the x we started with. So we can rewrite our initial function as:

y = x + 4,

knowing that it's still a function, just written in a way that's more conducive to graphing.

By the way, that last equation might look a bit familiar for those of you who've worked with slope-intercept format when graphing lines. It's y = mx + b. In fact, all first-power functions form a line when graphed, and all can be reworked to fit into slope-intercept format (though some get kind of unwieldy when you do). So now we see that there are two simple ways to graph a function: plotting several points and then connecting the dots, or working it into slope-intercept form and graphing it from there.

Functions are a lot of fun when you know what you're doing, and you can get into all sorts of complex graphs using higher power functions with the same concepts in mind. Just remember: a function is nothing more than a machine that turns x's into y's by following a set rule. Hopefully this journey helped you understand functions a bit better, and gave you the desire to get back in there and finish your homework!