How do you prove two languages are equivalent using the definition of acceptance?
I don't think you need to use induction, a simpler case analysis will suffice, along with a couple of central observations about the new DFA $f(M)$:


$q_{f}$ is the new start state,
$q_{f}$ is also an accepting state,
$\delta'$ is almost exactly the same as $\delta$ (in a very specific way of course), and
there's no transitions into $q_{f}$.


From this you can argue what happens on input $x$ in two cases: 


$x = \varepsilon$, and
$x$ is anything else.