In such a problem, we should consider an example. For instance, x = 2.
Considering the operations we have :
- F : square root ;
- G : multiply by a constant c ;
- H : reciprocal ;
We want to know which value of the constant "c" that would allow us to get
the same result no matter the order of operations applied. Since we have three operations, we get to have 6 possibilities (to be even more accurate 3! = 6) :
F - G - H : \(\frac{1}{c*\sqrt{2}}\)
F - H - G : \(\frac{c}{\sqrt{2}}\)
G - F - H : \(\frac{1}{\sqrt{2*c}}\)
G - H - F : \(\sqrt{\frac{1}{2*c}}\)
H - F - G : \(c*\sqrt{\frac{1}{2}}\)
H - G - F : \(\sqrt{\frac{c}{2}}\)
As you can see, there are instances where
the constant c is within a square root, or in the denominator of a fraction. Which means that zero and non-zero negative values can't be considered, therefore answers A, B and C are excluded.
Answer choice D is a fraction and seeing as the constant c can be either in the numerator or the denominator of the resulting fraction, c can either become \(\frac{1}{2}\) or 2. Which means we get different values everytime we change the order of the operations. Therefore, answer choice D is excluded as well.
Finally, we're left with one possible answer, E, which gives us \(c = 1\).
Hope that helped.