We see in the play that family life is in no way your conventional ideal of what we would call a nuclear family today. There are seldom any mentions of a "normal" family in the play.
We know that Lady Bracknell despises her husband in how she continuously puns at not having space for him at the dinner table, but that it would not matter to him whether to be there or not.
We also know that Gwendolyn hardly mentions her father but once (when she met Cecily) and then she shun him off in the conversation as someone not worthy talking about.
Lady Bracknell and Gwendolyn are the closest sample of "natural" family, and still what we see is that theirs is a relationship in which the mother controls the daughter for the purpose of grooming her for a rich marriage.
Ernest is a ward to Cecily Cardew. Since he was found inside a bag at Victoria Station by Cecily's rich grandfather, this man raised Ernest as his own child, and left him in charge of Cecily and the estate after he died. Here we also see a rare family circle composed of a man who was once found and abandoned taking care of a girl whose parents are also unmentioned in the play.
Finally, we have the actual Worthing family. We know that Miss Prism used to be Jack's governess when he was a baby and worked for his family. We know that she took him in a perambulator down to Victoria Station and "accidentally" left the baby, and not her manuscript, inside the back she checked in. We assume that Prism had gone to the Station to escape and publish her manuscript, leaving the baby behind. Yet, as she was confronted years later about the whereabouts of the baby, she could not answer. We do not know either what was the reaction of Jack's parents upon the loss of the baby. Another odd family practice in the play.
In the end, we know that Lady Bracknell was the sister of Jack's mother (and Jack's aunt), that Algernon is then Jack's brother, and that Jack's father's real name was Ernest and that he was a soldier at one point. Yet, still nothing else remains to be said about family in general: Other that, In the Importance of Being Earnest a family is as functional as its members. Hence, there are no functional families in the play.