Source: Revised GRE PDF 2nd Ed. Section 6; #23 (p. 93)

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# For all integers x, the function f

For all integers x, the function f is defined as follows. f(x) = x - 1 if x is even f(x) = x + 1 if x is odd If a and b are integers and f(a) + f(b) = a + b, which of the following statements must be true? a = b, a = -b, a + b is odd., Both a and b are even., Both a and b are odd.

### 2 Explanations

1

Phoebe Chua

What happens if a = b? Let's say a = b = odd. Then f(a)+f(b) = even + even = even.

Sep 8, 2015 • Comment

Cydney Seigerman, Magoosh Tutor

Great question! :D
Yes, you're completely right that if a = b and that number were odd, then f(a) + f(b) would be even. However, since f(x) = x+1 for odd numbers, then for two odd numbers, f(a) + f(b) = a + 1 + b +1 = a + b + 2. In the case that a = b, we could replace b with a to come up with the final expression:
f(a) + f(b) = 2a + 2.
This does not equal a + b, which, if a = b, would be 2a.
And 2a is not equal to 2a + 2.
That shows us that both a and b cannot be odd.
I hope this helps! :)

2

Chris Lele

Sep 27, 2012 • Comment

Francisco Olivas

Wonderful, now I get it! :-)

Ramlah Merchant

Would this method be correct- I simply plugged in a+1+b-1 = a+ b to test out an even and an odd number. since the 1s would cancel out it became very clear that one number would have to be odd and the other even for the left hand side to emerge as a+b.

Jonathan , Magoosh Tutor

Hi Ramiah,
The question asks with MUST be true, so if you plug in, you should be able to justify why the correct answer is correct for more than just the particular values you chose. If you thought "the only way to get a + b is if we have a + 1 + b -1 OR a - 1 + b + 1 so therefore a is odd and b is even or vice -versa" then you were correct :)