Metro train turns into private railway

We bought a 3 day Metro pass that gets us almost everywhere we want to go. On one day, we are traveling to Kawagoe. The route shown indicates that we start on a subway line that turns into another railway line as we enter Saitama. The journey continues on the original car we began in. The itinerary says we don’t change trains or switch stations. How does this work? Will we pay the difference when we get off the train in Saitama? Also, how does it work going back? Do we purchase a fare to the closest actual Metro station?

I’m going to answer my own question in case it’s helpful for anyone else. We got on the train and rode to our destination. After we got off we went to the station office next to the exit gate. We showed them our Metro pass. He charged us the amount from the last Metro station until the station where we exited.