Of course, there are lots of reasons. His shy good looks, that impishness, the fantastic curly hair! His wordy wit, his game-playing acumen (do not bother trying to win at Scrabble). He can retain an unnatural amount of information. He can remember almost everything (take note! he has forgotten things lately!).
But the quality of Jeff’s that I admire the most is his love for learning. He has an intellectual curiosity that is a perfect companion to my interest in what seems like any subject under the sun. However, unlike me, Jeff can sit down and learn a new programming language from a book and have it mastered in a couple weeks.
And he enjoys nothing more than sharing his knowledge with others.
The other night, a friend’s daughter needed help with her pre-Calculus homework. She called around 8:00pm and the homework was due the next day (!). So after all the dishes were done, the trash had been taken out, Dashiell’s lunch had been packed, and everyone else went to bed, Jeff reviewed her class notes and her assignment, and wrote her this email.
Let’s start with domain and range. These are just fancy words for what you put in and what you get out.
Think of it like this: If you have a job that pays $10/hr and you put in 8 hours, you get out $80. In function notation, this would be represented like f(x) = $10 * x, where f(x) is the fancy name for how much you get paid. So in this case, the “domain” [what you put in] is how many hours you work and the “range” [what you get out] is how much you get paid.
Now, mathematically, the “domain” of is function is all real numbers; however, for this example [maybe it’s not a good one for this reason] you could argue that the domain is really only x >= 0 because you can’t work negative hours.
Now, with the restrictions, they build on this concept and say, “okay, technically, what we’re representing in an f(x) notation is not a ‘true’ function, but it’s way easier to just write it this way.” When they ask “which values of x are not in the domain of the function,”what they mean is “so, okay, our formula metaphor is not perfect, where does it fall apart?” The restrictions are some “hints” of where formulas fall apart: 1) if you try to divide by 0 and 2) if you try to take the square root of a negative number.
So let’s take a look a real-world example: the strength of an electromagnetic field is roughly f(x) = k / (x^2), where k is something that doesn’t matter. If this is troublesome, just think of it as 5, f(x) = 5 / (x^2). So I know what you’re thinking, electromagnetic field strength? real-world example? Huh? Here’s where it gets real: Your cell phone emits electromagnetic radiation [range, or what you get out]. How much your brain gets depends on how far away it is [domain, or what you put in]. So how much radiation your brain gets “is a function of ” [a.k.a depends on] how far away from your head the phone is. Great, so we have a formula, and we kinda understand how it relates to real life, but how do I figure out what the domain is? This is where the restriction “hints” come into play.
So let’s start with the 2nd one: are we taking the square root of anything? 5 / (x^2) Nope, okay, so that hint didn’t help us. What about trying to divide by 0? Are by doing division? Yep. 5 / (x^2). Okay, so when would the bottom part be 0? x^2 = 0. We’ll that’s pretty obvious, when x = 0. So, the only value where our formula breaks down [a.k.a. not in the domain] is when x = 0. If we think about the cell phone-brain example, does this make sense? Sure, the cell phone can’t be a distance of 0 from our brain because at a distance of 0, the cell phone and our brain would be in the exact same place [ok, so sometimes, I think my brain is *in* my cell phone because of mobile google and wikipedia, but that’s completely different].
Ok, so let’s take a another example: maybe you’ve had a class where a teacher graded “on the curve,” meaning that they adjust everyone’s grades based on how some or all of the students did. The idea is that they use a “bell curve” to determine what an A is, what a B is, what a C is, etc. One of the things that they use to determine the width between the lowest A and the lowest B or the lowest B and the lowest C is called “standard deviation,” [again, a fancy term for width; those math people love their fancy words]. So the formula for width is based on based on the square root of a whole bunch of stuff like what all the individual scores were, what the average is, etc. We’re just going to call that whole ball of wax, x. That’s what we put into the function and what we get out is how much space is between an A and a B. In math notation, this would be f(x) = sqrt(x). So for grading on a curve, the range, or what’s an A, what’s a B, what’s a C, is a function of the domain, or how everyone scored. Ok, great, so what’s in the domain and what’s not? Well, let’s look at our hints: Are we dividing by anything? Nope, that hint isn’t going to help us. Ok, are we taking the square root of anything? Yes! Great, so when does our formula metaphor break down? Square roots of negative numbers. Ok, so x < 0 makes us take a square root of a negative number. Let's check this against the real-world example: So if 0 people take a test, the distance between an A and a B is 0. That's a little silly, but technically true, so that's ok. So what about x < 0? Well, can we have negative people take a test? Negative attitude doesn't count, so no, having -3 people take a history test doesn't make any sense. So we say that x < 0 is not in the domain because that makes our formula say ridiculous things. Make sense? Does this help? Let's work through the homework problems, 48, 49, and 50: 48. Which values of x are not in domain of f(x) = 3 / (x - 1)? Hint 1: Are we dividing by something? Yes! When does that thing equal 0? x - 1 = 0, so when x = 1, we are dividing by 0. 1 is not in the domain. Hint 2: Are we taking the square root of something? No! We're done. 1 is the only thing not in the domain. 49. Which values of x are not in the domain of f(x) = (3 - x) / (x + 5)? Hint 1: Are we dividing by something? Yes! When does that thing equal 0? x + 5 = 0, so when x = -5, we are dividing by 0. -5 is not in the domain. Hint 2: Are we taking the square root of something? No! We're done. -5 is the only thing not in the domain. 50. Which values of x are not in the domain of f(x) = (x^2 - 18) / (32 - x^2)? Hint 1: Are we dividing by something? Yes! When does that thing equal 0? 32 - x^2 = 0, so when x = +- sqrt(32) a.k.a +- 4 * sqrt(2) a.k.a. +- 5.6568... . +- 4 * sqrt(2) is not in the domain. Hint 2: Are we taking the square root of something? No! [So, yes, we're taking the sqrt of something to find the answer to hint 1, but we're not taking the sqrt of something in the function itself]. We're done. +- 4 * sqrt(2) are the only things not in the domain. By the way, you have an error on 41. f(-4) for x^2 - 13 should be (-4)^2 - 13, not - (4^2) - 13. So, (-4)^2 is 16. 16 - 13 = 3.