SAT Tip: Symbol Functions

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This tip on improving your SAT score was provided by Vivian Kerr at Veritas Prep.

A “function” describes a series of inputs and outputs such that each input provides exactly one output. The vertical line test will help you see whether a graph is a function: A function has no vertical lines that intersect the graph at more than one point. A typical function might look something like this: f(x) = 3x + 2. The input is “x” and the output is “f(x).” Don’t be intimidated by that “f(x)”—it essentially functions like the “y” variable you’re used to.

Sometimes functions will use made-up symbols. In these questions, the SAT test makers choose a strange symbol—something you might not have seen before—and gives it a specific definition. This type of question might seem scary, but they really require nothing more than simple substitution. Let’s look at some questions.

x ∑ y is defined by the equation x ∑ y = x2 – 1/y. What is the value of 4 ∑ 2?

Here, all you have to do is substitute  a 4 for x and 2 for y and then solve: 42 – 1/2

As you can see, we’re just doing simple substitution. Solving, we have:

16 – 1/2


Simple right?  One way the SAT can make this kind of question harder is to “chain” symbols like (4 ∑ 2) ∑ 3.  If you run into one of these, all you have to do is do the operation in the parenthesis first and then repeat the process again.

2. Let the function ⏀x be defined as ⏀x = (x + 3)(x – 3). If ⏀v = v + 3, what is one possible value for v?

(A)  9

(B)  6

(C)  4

(D)  3

(E)  0

If you are a little confused about where to start, remember that you can always use our TAC strategy—Test Answer Choices. Plug in the answer choices into the function to see which one will yield “v + 3.” If our answer is too large, we will be able to eliminate (A) and (B). If it’s too small, we’ll eliminate (D) and (E).

(C)  ⏀4 = (4 + 3)(4 – 3)

⏀4 = (7)(1) = 7

7 does equal 4 + 3, so (C) must be the correct answer. Let’s look at another question where we can use TAC:

3. Let the function fg be defined as fg = 2f-g. If h⋆12 = h, then h =

(A) 2

(B) 4

(C) 8

(D) 16

(E) 76

We’ll start by following the pattern of the symbols. If h⋆12 = h, then 2h-12 = h. Let’s start with choice (C):

If h = 8, then 28-12 = 8.

2-4 = 8, which isn’t true, so (C) cannot be correct. We need a LARGER number since our exponent is negative, so let’s move to choice (D).

If h = 16, then 216-12 = 16.

24 = 16

16 = 16

(D) is correct.

Look out for more opportunities to use the TAC strategy in symbol-function questions on test day, and always remember to start with choice (C). On the SAT, we know one of the 5 choices must be correct, so we can use that knowledge to our advantage.

Vivian Kerr has been teaching and tutoring in the Los Angeles area since 2005. She graduated from the University of Southern California, studied abroad in London, and has worked for several test-prep giants tutoring, writing content, and blogging about all things SAT, ACT, GRE, and GMAT.

For more SAT advice from Veritas Prep, watch “When Should You Guess on the SAT?”

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