SAT Newly Defined Functions

Correct Answer: D

Solution:

Tip: $x^{2} \geq 0.$
I. $a \oplus a = a-a+a^{2}=a^{2}$ . This can never be negative.

II. $(a+b) \oplus b = (a+b)-b+b^{2}=a+b-b+b^{2}=a+b^{2}$ .

If $a+b^{2}$ is negative, then

$\begin{array}{r c l} a+b^{2}&<&0\\ a&<&-b^{2} \end{array}$

This is possible. For example, if $a=-5$ and $b=2$ , we will get

$(-5+2) \oplus 2 = (-5+2)-2+2^{2}=-3-2+2^{2}=-5+4=-1$ ,

which is negative.

III. $(a+b) \oplus (a+b) = (a+b)-(a+b) + (a+b)^{2}=(a+b)^{2}$ . This can never be negative.

Only I and III can never be negative.

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Incorrect Choices:

(A) , (B) , (C) , and (E)
The solution above explains why these choices are wrong.