If φ is a function, does aφ 1 = 1 for all a? (1) 1φ a = 1 for all a. (

Question

If  φ  is a function, does  a φ 1 = 1  for all  a ?

(1)  1 φ a = 1  for all a.

(2)  a φ b = b φ a  for all  a  and  b .

Source:  Nova GMAT

Kinshook · Answer

Q: a φ 1 = 1?

S1:
 1 φ a = 1  for all a
NOT SUFFICIENT

S2:
 a φ b = b φ a  for all  a  and  b 
a φ 1 = 1 φ a (put b=1)
But the value of a φ 1 = 1 φ a is not known
NOT SUFFICIENT

S1&S2:
S1:  1 φ a = 1  for all a
S2: a φ 1 = 1 φ a
a φ 1 = 1 φ a = 1 
SUFFICIENT

IMO C

sayyamdeshpande · Answer

how do we know that b=1? What if b=2?

fmcgee1 · Answer

It works with 2 as well. Think of it like this (experts correct me if I'm wrong, but this is how I was thinking about it):

a φ 2 = 2 φ a

In order for this to be true,  a  can either any number, and the function, for instance, can be +, -, * to hold both sides true.

But now we at least know that a φ 1 = 1 φ a

Then, we take a look at statement 1, and since we know the a φ 1 = 1 φ a, we then know that 1 φ a = 1, which means that a φ 1=1.

Hope that was clear.

TestPrepUnlimited · Answer

Try substituting different operations to get a "feel" of how functions should work.

Statement 1:

1 φ a = 1 , we want to know if  a φ 1 = 1 = 1 φ a . As we know some operations are not commutative, for example, 1 / 2 is not equal to 2 / 1. Thus we have no reason to assume  1 φ a = a φ 1 = 1 , insufficient.

Statement 2:

No values mentioned. Insufficient.

Combined:

Now we have  1 φ a = a φ 1  from (2) and  1 φ a = 1 , thus  a φ 1 = 1 . Sufficient.

Ans: C

If φ is a function, does aφ 1 = 1 for all a? (1) 1φ a = 1 for all a. (

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If φ is a function, does aφ 1 = 1 for all a? (1) 1φ a = 1 for all a. (

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