For all real numbers a and b, where ab\neq0,… | GMAT Sprint

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PS · Problem Solving

705-805

Functions

For all real numbers $a$ and $b$ , where $ab\neq{0}$ , let $a@b=a^b$ . Then which of the following MUST be true?
I. $a@b=b@a$
II. $(-a)@(-a) =\frac{(-1)^{-a}}{a^a}$
III. [m](a@b)@c=a@(b@c)[m]

Answer Choices

Correct answer marked below

I only

II only

Correct

III only

I and II only

II and III only

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Explanation

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Official Explanation

Official

a@b=b@a

→

a@b=a^b

and

b@a=b^a

; these 2 expressions are not always equal:

2^3\neq{3^2}

.

II.

(-a)@(-a)=\frac{(-1)^{-a}}{a^a}

→

(-a)@(-a)=(-a)^{-a}=(-1\cdot a)^{-a}=(-1)^{-a}\cdot a^{-a}=\frac{(-1)^{-a}}{a^a}

; this matches the given expression, so the statement is always true.

III.

(a@b)@c=a@(b@c)

→

(a@b)@c=(a^b)^c=a^{bc}

and

a@(b@c)=a^{(b^c)}=a^{b^c}

; these 2 expressions are not always equal:

2^{2\cdot 3}=2^6=64\neq 2^{2^3}=2^8=256

.…

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What this question tests

What it tests

Your ability to evaluate and compose functions, including unusual defined operations.

Common trap

Applying function operations in the wrong order or ignoring domain restrictions.

For all real numbers aaa and bbb, where ab≠0ab\neq{0}ab=0, let a@b=aba@b=a^ba@b=ab. Then which of the following MUST be true?I. a@b=b@aa@b=b@aa@b=b@aII. (−a)@(−a)=(−1)−aaa(-a)@(-a) =\frac{(-1)^{-a}}{a^a}(−a)@(−a)=aa(−1)−a​III. [m](a@b)@c=a@(b@c)[m]