This thread is NOT useful to anyone preparing for the SAT. I was working on the super difficult problems in “Math SAT Prep 800” to improve my skills as a tutor or whatever. I couldn’t do this problem and it doesn’t make sense to me. The solution states that f(x^2) = f(x) * f(x): I believe that is clearly wrong.
Is the solution correct? Is the problem solvable? Does anyone know how to solve it.
Problem 4, Test 4 p. 61
f(x^2) = f(x) + 2
f(xy) = f(x)*f(y)
Let f be a function that satisfies both of the above properties for any real number x and y. Then, which of the following could be the value of f(1001)?
a) -1 b) -2 c) 3 d) 102 e) 1003
Solution:
Since f(x^2) = f(x*x) = f(x) * f(x), we have
f(x^2) = f(x)^2 = f(x) + 2 or
f(x)^2 – f(x) – 2 = 0
Factoring this in terms of f(x) gives us
(f(x) – 2) (f(x) +1) = 0
which yields f(x) = 2 or f(x) = -1. This result suggests that f(x) is always either 2 or -1. Therefore, -1 is a possible value of f(1002).
Yes, technically that is correct. Let x = y in the second equation.
However, it becomes clear that this is a poor question and wasn’t proofread. The question says that f(x^2) = f(x) + 2 for any real number x and y, so it must hold when x = 0. But then f(0) = f(0) + 2, or 0 = 2. Furthermore, if f(x) ∈ {-1, 2} for all x, then no choice of values for f(x) satisfies the given properties.
It is not a garbage prep book, although there are many. It has maybe the hardest problems anywhere. It is difficult making up problems. The “Prep 800” book made problems that were fairly realistic, but extremely hard and not just copies of real problems with minor changes. It is useful for my purposes, as I can do all the problems in the standard books. I think it is useful book for some students and fills a niche.
@sattut That is not a valid counterexample. The question states, let f(x) be a function that satisfies both of the above properties for any real number x and y, so it assumes that f satisfies f(xy) = f(x)f(y). A counterexample to your claim would be a function that somehow satisfied f(xy) = f(x)f(y) for all real x,y but fails to satisfy f(x^2) = f(x)^2. In this case, f(x) = 3x+2 doesn’t satisfy the first condition.
I agree; it is difficult to write good test problems. I’ve written many myself for other purposes (e.g. math contest proposals). But problems should be carefully revised before publishing, as you have to take lots of factors into account (well-defined, problem difficulty, amount of brute-forcing required, distribution of topics, etc.), and it is a good idea for others to test-solve your problems.
In this case, the author should have shown that there exists a function f that satisfied the requirements and f(1001) = -1. The fact that this mistake actually exists in their solution doesn’t convince me that these problems are good or realistic.
It is a valid counterexample, because the author was arguing that f(x^2) = (f(x))^2 in general, which it clearly does not. The solution didn’t state that property was true for that particular weird function, but in general.
The whole problem doesn’t make sense. There is no way a function could satisfy those conditions.
The book has 12 authors, and most of the problems are not like that. Obviously, they should have checked each others work better.
@sattut not sure where you got the “in general” part. It definitely holds when f satisfies the second property, although a clearer way to reword the solution (and this is what the author likely meant) would be:
"Let f(x) be a function that satisfies the above properties for all real x and y. Then
Are your “purposes” to tutor students for the SAT? If so, I don’t understand why going beyond the difficulty of problems on the actual SAT exam is helpful. Can you elaborate?
While I am on the record (many times) favoring real materials, I do see the use of a source of hard problems if they are well-written and just barely over the edge of SAT difficulty. And I would not condemn a book based on one problem. Yes, this one is mess, but that does not make the whole book “garbage”. I can tell you that if you write many problems, an occasional goof will slip by.
But I do want to re-emphasize that an important part of SAT practice includes self-calibration and time management. So treat these other books as work-books and not as predictors, and keep the real tests as the major focus of your prep work.
Yeh, I agree that an important part of preparation should be real SATs. Also, books like this are useless predicting scores or time management: for that take whole real SATs.
I disagree with some here, and would recommend this book and the similar “Math SAT 800” and “Crush the Math SAT” to students with scores in the 700s and time to practice. At least, you will not be intimidated by the hardest problems on the real test if you practice with these.
Yes, we all make mistakes, and we should all acknowledge that. Ideally though, such mistakes should be caught before submitting a final version. I might have been a bit too harsh with classifying the book as “garbage” based on one garbage problem, but if it gets to the point where there are multiple bad problems, then I would almost certainly recommend not buying that book.
I have gone through that book thoroughly; of 500 problems, there are at most 5 with errors, and that is by far the worst error. Also, they do a better job of making realistic problems without being close to copies of real problems than most books do.
Per post #5:
" The book has 12 authors, and most of the problems are not like that. Obviously, they should have checked each others work better. "
Newsflash, many student solutions manuals for college textbooks are done by different authors than those who did the associated textbooks. We continually get many errors, sometimes tens of errors, in student solutions manuals even that are on their fifth or higher editions.
And with 12 authors, it is doubtful anyone but a single chief author or editor went through every problem.
I think you are expecting too much from real life, when real life is made up of humans interacting.
It costs money for reviewing and editing, and even as “specialized” as an SAT math review book likely means that an editor with dubious math knowledge is doing the final check.
I know professors who have authored textbooks, and they use grad students to check their work. Most textbook editors do not have math or science or history or whatever degrees, they have English degrees.
There are plenty of college students or people in India who can score close to 800 on the math SAT who could go over all the problems cheaply. There really isn’t an excuse for errors like this. You also rarely the see errors in Barrons books or the Pwn book. However, I wouldn’t condemn this book for a few errors.
After working through books like that, the real level 5 problems seem easy to me, so I would see that as a benefit. I think there would be a value in a book of just level 4 and 5 problems, rather than the books of mostly harder than 5 problems that exist. I also think there would be value to a book of hard SAT writing problems.
This thread is indeed not useful for anyone working on the SAT. And neither is … The book, if it is the Eiblud book.
It is not garbage for anyone who wants to refresh a bit of his math knowledge, but it IS garbage in terms of SAT prep as it falls in the Chung category of gross irrelevance.
To be fair, OP was using this for working on his own skills. That’s fine and it does not waste any of the student’s time.
But for a student who wants an 800, I still feel that there are enough real tests to get there.
I can’t imagine a student with 800-level talent who has mastered even just the 10 in the blue book, to the point where they can explain any problem, who then goes on to under-achieve. I guess you can throw in the real tests from the on-line course if you want, but I think even that is more than it takes. With my students and also with my own children, I have not even ordered the online course.
And if you tutor, ask yourself what percent of your students actually reach the point where they have completely mastered the entire blue book. But again, by “mastered” I mean that you can point to any problem and they can show you how they do it now, how they did it the first time they saw it, what math concepts it is testing – essentially, they could step in and tutor someone else! Working to achieve that level of mastery with the authentic material to me feels like a better use of student time than reaching out for artificially harder test material.
By the way, I think this is even more true for verbal than it is for math.
I don’t really agree about just real tests for 800. When tutoring a student at 700+, I have them do the level 4 and 5 problems in the Blue Book, and maybe also the older red and white real SAT books and online real SATs. There are not all that many hard real problems, so I have sometimes used the Eiblum book. It is more practical for tutoring and easier than this “Prep 800” and the “Crush” book. There are other prep books I sometimes use with weaker students.
When I am tutoring, I do redos on real SAT problems the student gets wrong. However, most 700+ students understand my explanations and get redos right, so I don’t see the point in focusing further on problems they can do.
The problem I listed makes the “Prep 800” book look really bad. I agree that it and Dr. Chung’s have some irrelevant material, but IMO neither is useless.
IMO the superhard books are useful for 700+ students who have the time, but I would recommend working through real SATs, Pwn, and probably some other materials first.
I was working on “Prep 800” partly as a challenge, and partly to improve my skills tutoring 700+ student. Also, I get 36 on the math SAT and easily 800 on the level 2 math SAT II, but I have scored 790, 790, and 760 when I have taken the math SAT I at tutoring centers, and I would like to get 800.
(I think that’s a winky face)