Greetings again, my function wisdom-seekers. Last week we went over Part I and Part II of this mini-series. Today’s topic of focus will be symbol functions: the lovely little questions where the SAT defines an entirely new, random symbol as an operation and has you work with it in some way. These problems can be downright bizarre at first since they are highly SAT-specific, but once you get to know them they aren’t so bad at all. Let’s take a look…

For all positive integers x, y, and z, **x** y** z** is defined as xy + yz – xz. What is the value of **7** 6 **12** ?

First, re-read the first part of the question, where the SAT defines the new symbol function for you. I highly recommend writing it out for yourself, so that you get it into your brain that way.

So, okay, for any three positive integers, the notation **x**y**z** means that we multiply the x and y together, add that to the product of y and z, and subtract the product of x and z. **x**y**z** = xy + yz – xz.

Then, re-read the second part of the question, to see what the SAT is asking you to find, and do whatever you need to find it.

In this case, they’re asking you for the value of **7**6**12** . To find that value, simply plug 7 for x, 6 for 7 and 12 for z into the function and evaluate. So, you’re going to get (7)(6) + (6)(12) – (7)(12), or 42 + 72 – 84 = 30.

A slightly trickier variation on this might require you to solve for a variable. For example…

For all positive integers x, y, and z, **x** y** z** is defined as xy + yz – xz. If **x**5**10** = 45, then what is the value of x?

As before, re-read/re-write the first part of the question, so that you understand what this function means as defined in this question. So, we know that for the three positive integers x, 5 and 10, **x**5**10** = 45.

Then, re-read the second part to see what you’re solving for and do whatever you need to solve for that.

In this case, you’re looking for the value of x. To figure that out, plug the numbers you already know into the equation to solve for the unknown. So, x(5) + (5)(10) – x(10) = 45. That means that 5x + 50 – 10x = 45. Combining like terms, we get -5x = -5, which means that x = 1. Problem solved!

An even slightly trickier version of this kind of problem might involve a function that’s a bit more algebraically complicated, e.g. J x, y J = (x + y) / (x-y), but the solving process would be the same as the one discussed above.

The last big symbol function question type is the Roman number one: the type of question where you’re given a symbol function and then asked which of the following statements (roman numerals I, II, and III) could be true, or must be true.

For example:

a @ b is defined as ab – (a+b)b for all real values of a and b. If a and b must be positive integers, which of the following could be equal to 0?

I. a @ b

II. a @ 2b

III. 2a @ b

These are tricky. In lieu of solving out this particular one, I’ll give you a few general tips for these. First of all, check to see whether the question is asking whether it “could” be true or “must” be true – if it’s “could,” all you need is one example to prove it; if it’s “must,” all you need is one example to disprove it. Write the function out, and think through each of the three statements and whether there’s any way it could (or could NOT) be true given the setup of the function and the restrictions defined in the question. Also, try strategically picking and plugging in numbers for variables in each case; this can help you see patterns and understand the possibilities and how the function works.

And that’s all we’ve got for the main types of symbol function problems! Tune in next time for some insights on graph and table – based function problems. Happy functioning until then!

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*Alice Rothman-Hicks is a Veritas Prep SAT 2400 instructor. Since graduating from Columbia University (Magna Cum Laude, Phi Beta Kappa), Alice has been teaching and tutoring test prep, helping students achieve their own academic successes. She scored a 2350 on the SAT.*