At work and at home, responsibility is chosen by you: you take up a responsibility, thus you choose to do it.
Responsibility is not what is given to you, it is what you choose to be responsible for.
Responsibility cannot be given; it can only be taken up.
What’s given to you is a job, a duty, a requirement, an obligation, or an onus. Whether you take it up as a responsibility is a choice left to you.