Problem Solving

If we multiply our positive number x by a negative constant c, then take the square root, we get something undefined. And if we multiply x by 0, then take a reciprocal, we get something undefined. So if we're supposed to be able to apply these functions in any order, just to get something that makes sense every time, c will need to be positive. But if c is 1/2, then if we multiply x by 1/2, then take the square root, we end up with x/√2, whereas if we first square root, then multiply by 1/2, we end up with √x/2, and these are different in value. So the only remaining possibility is that c=1, which makes sense as an answer, since multiplying by 1 doesn't change anything.

AAPL wrote:Manhattan Prep

Operation F means "take the square root," operation G means "multiply by constant c," and operation H means "take the reciprocal." For which value of c is the result of applying the three operations to any positive x the same for all of the possible orders in which the operations are applied?

A. -1

B. \(-\frac{1}{2}\)

C. 0

D. \(\frac{1}{2}\)

E. 1

OA E

We see that the problem is saying, for example, that H(G(F(x)) = H(F(G(x)). Expanding both sides of the equation, we have:

1/(c√x) = 1/√(cx)

c√x = √(cx) = (√c)(√x)

Since √x is positive, we can divide each side by √x:

c = √c

Of the given answer choices, only c = 0 and c = 1 satisfy √c = c. However, c cannot equal 0 because otherwise, the expression 1/(c√x) (and some of the other expressions obtained by applying the operations in different orders) is undefined. Thus, it must be true that c = 1.

Answer: E