18. For all positive integers p and q, let the function p♮q be defined as p♮q = 2^p-q. If g♮12 = g, then g =
     
    (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
2⁴= 16
BOOM GOES THE DYNAMITE.

(D) is our answer.