Here’s another section for the mathematics textbook I’m not writing, Tropic of Calculus.

One of the concepts pre-Calc students seem to wrestle with is Domain and Range. Here are some key points to make sure you understand.

First, let’s make sure you have a good, casual understanding of what domain and range are. For some function, y = f(x), the “domain” is the span of all possible X values for that function, while “range” is the span of all allowable Y values for the function.

One way to think about it is to look at a graph of the function. You look at the X axis as if it was a number line and you ask yourself, “Self, as I traverse across this number line, does the function exist somewhere (above or below the axis) no matter where I am on the number line?” When you’re looking at the X axis, the answer you get is the domain of that function, and when you’re looking at the Y axis, that’s the range. This is a good first step for helping you wrap your head around domain and range.

And here’s another way to think about it. Let’s imagine you had infinite time, and you made a list of every possible coordinate point the graph of the function passed through. Yeah, that’s an infinite list, but still, let’s pretend. When you were done, you could summarize the results by looking at every single X coordinate you listed. Does that list cover every possible X value? Or does it skip some numbers? That’s the Domain of the function. And then let’s do the same for the list of all possible Y coordinates. Does the list cover every possible Y value, or does it miss a few numbers? That’s the Range of the function.

But eventually you want to start thinking about domain and range simply by looking at the equation itself.

For domain, ask yourself “Is there any X value I am _not_ allowed to plug into this equation? Can I plug in stupidly small negative values, like negative one million? Can I plug in zero? Can I plug in stupidly large positive values, like positive one million?” Equations tend to be fairly flexible about inputs, and mostly the answer is “Oh sure, I can plug in any X I want. So my domain is “all real numbers”.”

At this stage of pre-Calc, there are two big “gotchas” about domain to make sure you look for.

Domain Gotcha #1: Even roots
Let’s say you have a function like:
y = (x – 7)^(½)
(That’s y = square root (x – 7))
The “gotcha” is that you cannot take the square root of a negative number. So you have to make sure the entire expression under the square root is never negative. Said another way:
x – 7 ≥ 0
And you can simplify that:
x ≥ 7
And so there’s your domain for this function. You can plug in any X, as long as it is greater than or equal to 7.
It should be pointed out that this “gotcha” is for any even root. (Square root, fourth root, sixth root, etc.) It’s totally possible to take the odd root of a negative number.

Domain Gotcha #2: Rational functions
If you have a function that is a rational function (it looks like a big nasty fraction, with variables in the denominator), that also presents a possible red flag.
Let’s say your function looks like:
y = 1 / (x + 5)
The “gotcha” is that you cannot divide by zero. So the entire denominator of your big nasty fraction cannot be zero.
x + 5 ≠ 0
And simplified:
x ≠ -5
And there’s your domain: all real numbers, except x = – 5.

Later in pre-Calc, there are other gotchas you might add to the list. For instance, not being able to take the log of a negative number, but for now, these two areas are the big ones to make sure to cover. You can’t take the even root of a negative number, and you can’t divide by zero.

And now the bad news. There’s not a similar well-defined set of “gotchas” for Range. You have to look at the equation and puzzle it out. For instance, let’s look at our previous two functions:
y = (x – 7)^(½)

After pondering for a while, you might realize that when you take the square root of something, you’ll never get back a negative result. And there’s your range: y ≥ 0.

For this function:
y = 1 / (x + 5)

A rational function will equal zero only when the numerator of the fraction equals zero. Since the numerator of this fraction is a constant 1, Y will never equal 0. And there’s your range, all real numbers, except y = 0.

But you really have to spend some time puzzling these out. When in doubt, look back at the graph of the function to double-check your beliefs about the range of the function.

Here’s a chapter section from the “math in the vernacular” textbook that I’m not writing, “Tropic of Calculus”.

(Ed note: The previous section would have covered the definition of functions and the use of the Vertical Line Test)

5.1 Function Notation

If you were raised in the United States, you probably have a first name, middle name, and last name. But unless you’re in trouble and your mother is lecturing you, people probably don’t call you by your full name very often. Instead of being called “James Tiberius Kirk”, your friends probably call you “James”, “Jim”, or even “Captain”. It’s just more convenient to refer to people with a nickname or a shortened version of their name, rather than using their full name every time you want to talk about them.

The same is true of functions. (Remember those? In the previous section we talked about the specific type of mathematical equation we call “functions”.)
You might have a function with a rather lengthy “full name”. For instance:

y = 404 x^7 + 1690 x^5 + 43 x^4 + 15 x^3 + 2020 x^2 + 5133 x + 19

If you had to repeat that entire thing every single time you wanted to talk about that function, you’d be exhausted before your class was even half over. So mathematicians have developed a “nickname” for specific functions, a shorthand for referring to functions.

Using the function from above, the “nickname” looks like this:

f(x) = 404 x^7 + 1690 x^5 + 43 x^4 + 15 x^3 + 2020 x^2 + 5133 x + 19

f(x)? What? Okay, let’s break it down.

This does not mean “f times x”. I know it looks like it does. It’s confusing that way. On behalf of mathematicians everywhere, I apologize.
Nor is ” f ” in ” f(x) ” shorthand for the word “function”. Mathematicians are just as likely to use ” g(x) “, or ” h(x) “, ” s(t) “, ” v(t) “,… the list goes on and on.

So what does it mean? It’s just a nickname for the function. If it helps, think of the ” f ” in ” f(x) ” as standing for “Frank”, or the ” g ” in ” g(x) ” standing for “George”.

What we’re trying to say is “Here’s a function. We’re gonna call it ” f “.”
And the ” (x) ” provides just a little extra information. It tells you that the independent variable for our function is an ” x “.

Here’s the “Math-to-English™” translation of a couple of examples:

Math: f(x) = x^2 + 4 x + 4
English: See that equation over there? It’s a function. And we’re going to call the function ” f “. The independent variable in the function is ” x “. And the full equation for the function is ” x^2 + 4 x + 4 ”

Math: g(x) = 1/x
English: Here’s another functions. This one is called ” g “. Yo, g. The independent variable in this function is ” x “. And the full expression for the function is ” 1/x “.

Math: s(t) = -16 t^2
English: I’m calling this function ” s “. The independent variable in this function is ” t “. And the full expression for the function is ” -16 t^2 “.

And if you’re reading this aloud, ” f(x) “, ” g(x) “, ” s(t) ” is read as “f of x”, “g of x” and “s of t”.

5.1a Independent and Dependent Variables

As we said above, for some function…
f(x) = x^2 + 4 x + 4
… ” x ” is the independent variable in the function.

Just to recap, the “independent variable” is the same thing as the “input variable”. It stands for some number you might be plugging into the equation. And if you’re graphing the function, this will be the horizontal axis for your graph.

So if ” x ” is the input, the entire ” f(x) ” is the output. Think of it like this: you plug some x into the function, do some arithmetic, and the ending result is your output. The “output variable” is the same as the “dependent variable”. And if you’re graphing the function, this will be the vertical axis for your graph.

We can use function notation to talk about the plugging in specific values into a function.

s(t) = -16 t^2

Let’s say we want to plug a ” 1 ” into this function. We’d just replace the ” t ” in the function with a ” 1 “, right? We do the same thing in the function notation.

s(1) = -16 * 1^2 = -16
1 was the input, – 16 is the output.

Similarly,

s(2) = -16 * 2^2 = - 64
2 was the input, – 64 is the output.

s(3) = -16 * 3^2 = - 144
3 was the input, – 144 is the output.

If we were graphing this function, we know the function would pass through the following coordinates:
(1, -16), (2, -64), (3, -144)

We can even use function notation to input something more than plain numbers into a function. For instance:

f(x) = x^2 + 5 x + 6

Let’s say we wanted to plug some constant into f(x). In this example, our input is ” π “, which is the number 3.14159… (it goes on and on and on…)

f(π) = π^2 + 5 π + 6 ≅31.5776

We could also plug in a “placeholder constant” that doesn’t have a specific value just yet, like ” n “.

f(n) = n^2 + 5 n + 6

We can even plug in slightly more complicated values, like ” 2m ”

f(2m) = (2m)^2 + 5(2m) + 6
= 4 m^2 + 10 m + 6

Or plugging in ” x + h ”
f(x + h) = (x + h)^2 + 5(x + h) + 6
= x^2 + 2xh + h^2 + 5x + 5h + 6

Here are some examples for you to try on your own. You’ll want to get comfortable with this topic, because we’re going to be building on it a lot in later sections.

(Ed note: domain and range, composite functions, inverse functions and transformations of functions)