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
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.