Count the functions that have an entry with a specific structure
If you want the function to have the property for one input, say $x$, which is fixed ahead of time, you just focus on that output vector
$$
f(x)=f(x_1,\ldots,x_n)=(z_1,\ldots,z_{2n})
$$
and the condition $z_i=z_{i+n}$ holds with probability

$$P(given ~row ~is ~good)=\frac{2^{n}}{2^{2n}}=2^{-n},$$
since in this case you don't care what is happening in the other rows. 

If you want this property to hold only for your selected input, this happens with probability

$$P(given~ row ~is ~good) (1-P(given ~row ~is ~good))^{2^n-1}=2^{-n}[1-2^{-n}]^{2^{n}-1}$$ 
 which can be estimated for large $n$ from above and below by using
$$
1-k\varepsilon+\frac{k(k-1)}{2}&lt;[1-\varepsilon]^k&lt;1-k\varepsilon
$$
where $\varepsilon$ is small and positive.