If x ≠ 0 and x ≠ 1, and if x is replaced by 1/x…
Let’s start by substituting 1/x for x. This gives us:
Even though our answer does not match any of the answer choices, it is really similar to choice A. In fact they are the same. We can demonstrate this by factoring out a “-1” from the denominator of our fraction. So we can say:
1 – x = -(-1 + x) = -(x – 1)
Plugging this back into the fraction we have:
But, because this entire expression is being squared, the negative would turn into a positive, and the end result would be the same as (1+x/x-1)^2.
Note: If this is difficult to see, let’s do the same thing with an integer. Let’s say we have 2. Well we could say the following:
(-2)^2 = (2)^2
4 = 4
Notice that because we squared the -2, it ended equaling 4. Although the expression in the problem above seems more complicated, the math behind what we did is exactly the same.