Function whose domain is strictly included in its range Hi I'm doing some exercices on reductions and one of them is like this: 



My problem rather than doing the reduction is understanding it. For the positive case I want that if Mx(x) never stops then p's domain has to be strictly included in q's domain which has to be stricly included in p's range. Am I wrong? Therefore Dom(p) has to be stricly included in Range(p), right?

Let's forget about reductions for a moment, my problem here is: is there any function $f(x)$ whose domain is strictly included in its range?

We would want to have $Dom(f(x)) = \{1\}$ and $Range(f(x)) = \{1, 2\}$ That would mean that 1's image has to be 1 AND 2 :/

What am I doing wrong?

Thanks for your time and have a nice day! :D
Without getting to the technicality of codomain/range/image and surjective functions, the property you ask for can surely happen. For instance, 

$$f(x) = 2x$$

defined over the domain interval $[0,1]$, has the image $[0,2]$. This should not happen over finite domains.