I heard this yesterday from a customer-service agent.

Problems:
1. The question ended on a preposition.
2. The word “back” is a poor substitute for what the agent should have said.

Explanation:
The full question was “Is this a good number to reach you back at?”

The agent wanted to confirm that the telephone number that she had would be a valid number to call in the event of getting disconnected during the current call.

Removing the preposition “at” from the end of the question requires rewording “to reach you … at?” as “at which to reach you …?”; that solves the first problem.

The second problem comes from laziness on the part of the agent. This is obvious when one realizes that “back” was a substitute for “in the event that we get disconnected”. This gives us the solution to the second problem.

Solution:
“… good number at which to reach you in the event that we get disconnected?”