An SAT Question on Functions

Date: 08/23/98 at 12:13:01
From: rachel
Subject: SAT Question

I was taking an SAT prep test on the computer from a program that I 
have. I ran across a question that I was completely unable to answer: 

Let the "tricate" of a number x be defined as one-third of the smallest 
multiple of 3 greater then x. If the tricate of z is 3, which of the 
following could be the value of z?

   A)  2
   B)  5
   C)  7
   D)  9
   E) 11

The question comes with an explanation of how to get the answer but I 
am still confused. What is a tricate? The explanation does not help at 
all:

Read carefully. If the tricate of z is 3, that means that one-third of 
the smallest multiple of 3 greater than z will be 9 only if 6 < z < 9.  
Answer choice (C) is the only number in their range so it must be 
correct.  

Now here is my problem. I can see how they reached 7, because it is 
greater then 6, and less then 9, but how did they come up with 6, how 
did they come up with 9, and how does this relate to x? Again, what is 
a tricate? I think if I first off knew what a tricate was, then I 
would be able to solve this problem.

Date: 08/24/98 at 12:38:51
From: Doctor Peterson
Subject: Re: SAT Question

Hi, Rachel. 

I think you may be confused by the word "tricate". The fact is, they 
have told you what it is, and you know better than I did coming in to 
this problem what it means. That's because what they are doing is 
defining a made-up term for you (one that no one in the history of math 
has ever used, as far as I know), and asking you to apply it. 
Mathematicians are used to defining new words, sometimes just for use 
in one article, but that idea may be new to you.

Maybe you will be a little more comfortable with calling it a function:

   t(x) = 1/3 of the smallest multiple of 3 greater than x

What in the world does that mean? Let's try to sketch a graph of this 
function. For x = 0, t(x) is 1, since the smallest multiple of 3 
greater than 0 is 3, and 1/3 of that is 1. In fact, until x reaches 3, 
t(x) will remain the same. Do you see why? Then for any x from 3 to 6 
(excluding 6 itself), t(x) will be 2, which is 1/3 of 6, the next 
multiple of 3. So the graph will look like this:

    |
   4+                          *
    |
   3+                 *--------o
    |
   2+        *--------o
    |
   1*--------o
    |
   0+--+--+--+--+--+--+--+--+--+--
    0  1  2  3  4  5  6  7  8  9

So the numbers z for which t(z) = 3 are those for which:

   6 <= z < 9

Does that help?

- Doctor Peterson, The Math Forum
Check out our web site! http://mathforum.org/dr.math/