Operation F means “take the square root,” operation G me

BTGmoderatorLU wrote: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. –1/2
C. 0
D. 1/2
E. 1

Let x be a positive constant. Let’s apply the operations F, G and H to x in various orders:

F(G(H(x))) = F(G(1/x)) = F(c/x) = √(c/x)

H(G(F(x))) = H(G(√x)) = H(c√x) = 1/(c√x)

Since we are looking for a value of c which produces the same result when the operations are applied in any order, let’s equate these two values:

√(c/x) = 1/(c√x)

(√c)/(√x) = 1/(c√x)

Since x is positive, we can multiply each side of this equation by √x:

√c = 1/c

c√c = 1

Let’s square each side:

c^3 = 1

c = 1

Answer: E