- For all positive integers
pandq, let the functionp♮qbe defined asp♮q= 2^{p-q}. Ifg♮12 =g, theng=

(A) 2

(B) 4

(C) 8

(D) 16

(E) 76

Answer below.

The cardinal rule of function questions is this: FOLLOW THE DIRECTIONS EXACTLY.

If *p*♮*q* = 2^{p-q}, then *g*♮12 = 2^{g-12}. We know that’s equal to *g*. Here’s what we have so far:

2^{g-12}= *g*

From here, we COULD start listing powers of 2 to see which ones might work, *or* we could take a shortcut and **BACKSOLVE** because we know one of the 5 answer choices has to be right. I vote the latter. As always when backsolving, start with (C). That says:

2^{8-12}= 8

2^{-4}= 8

**NOPE**.

Note that 8 is too small…it creates a negative exponent on the left of our equation. So let’s move towards (D) and see what happens:

2^{16-12}= 16

2^{4}= 16

**BOOM GOES THE DYNAMITE**.

(D) is our answer.

There is no math solution…?

Of course there is…but my purpose in this post is to show you another way. Note that the math solution (which I’ll let you find) involves math that’s not necessary on the SAT. I made this problem up to illustrate a point, not simulate the SAT exactly.

Mike, if g= 2^(g-12), then does that mean that p= 2^(p-q)? and q= 2^(p-q)?

Nope! It just means that for the one constant g, that you’re going to solve for, that happens to be true.

Would the SAT ever have a question like this as a grid in?

No, probably not.