Approach #1: Create equivalent fractions
Key property: We can create equivalent fractions by taking a fraction and multiplying the numerator and denominator by the same value

When I scan the answer choices, I see that answer choice E looks very similar to the given fraction \(\frac{1}{\frac{1}{x}+1}\)
Notice that, if we take \(\frac{1}{\frac{1}{x}+1}\).....
.... And multiply numerator and denominator by 2, we get the equivalent fraction \(\frac{2}{\frac{2}{x}+2}\)

In other words \(\frac{1}{\frac{1}{x}+1} = \frac{2}{\frac{2}{x}+2}\)

So, if \(\frac{1}{\frac{1}{x}+1}=k\), then it must also be true that \(\frac{2}{\frac{2}{x}+2} = k\)

Answer: E

Approach #2: Test a value of x

If \(x = 1\), then we get: \(\frac{1}{\frac{1}{x}+1} = \frac{1}{\frac{1}{1}+1} = \frac{1}{2}\)
So, it must be the case that the correct answer also evaluates to \(\frac{1}{2}\), when \(x = 1\)

We'll now plug \(x = 1\) into each answer choice to see which one evaluates to \(\frac{1}{2}\)

A. \(\frac{1}{\frac{2}{1}+2}= \frac{1}{4}\). No good. We need \(\frac{1}{2}\). Eliminate

B. \(\frac{2}{\frac{1}{2(2)}+1}= \frac{8}{5}\). No good. We need \(\frac{1}{2}\). Eliminate

C. \(\frac{2}{\frac{2}{1}+1}= \frac{2}{3}\). No good. We need \(\frac{1}{2}\). Eliminate

D. \(\frac{2}{\frac{1}{2(2)}+\frac{1}{2}}= \frac{8}{3}\). No good. We need \(\frac{1}{2}\). Eliminate

E. \(\frac{2}{\frac{2}{1}+2}= \frac{1}{2}\) PERFECT

Answer: E