Functions – – Ugh!

A function is defined for x and y such that f(x,y) = -2xy + y + x – 4. So, for x=2 and y=3, f(2,3)= -2x2x3+3+2-4 = -12+1 = -11. If x and y are to be chosen such that f(x,y) = f(y,x), then which of the following restrictions must be placed on x and y?

A) x>0 and y>0
B) x<0 and y<0
C) x = y
D) No restrictions are needed

Hmm…

The thing to recognize here is that nothing happens to y in the function definition that doesn’t also happen to x. Both get multiplied by –2 and each other, and then both get added.

From that, choice D is looking pretty good, but let’s try a counterexample that will eliminate A, B, and C all in one fell swoop. Let’s say x = 3 and y = –2. If f(3, –2) = f(–2, 3), then we know choices A, B, and C, don’t need to be true.

f(3, –2) = –2(3)(–2) + (–2) + 3 – 4
f(3, –2) = 12 – 2 + 3 – 4
f(3, –2) = 9

f(–2, 3) = –2(–2)(3) + 3 + (–2) – 4
f(–2, 3) = 12 + 3 – 2 – 4
f(–2, 3) = 9

Yep, turns out that f(x, y) = f(yx) when one of them is positive and one of them is negative, which means none of the proposed restrictions are necessary.