What was the significance of Alex lying about her dad's death?

So Alex begins by telling Dan that her father died of a heart attack when she was seven years old, only to take it back a moment later saying, "He's alive and well and living in Phoenix," and laughs about it.

Later, in the movie, Dan sneaks into her apartment and finds the obituary that mentions her dad did indeed die of a heart attack. The music begins as if there is some great revelation to the fact she was originally telling the truth about her dad's death.

So first of all, what was the significance behind this? It's like that's all we learn from the scene where Dan gets into her apartment. Yet, it's like we learn something about Alex's psychological state, but what?

Was it just to show she had mental issues? It doesn't say much that she lied about this to me. What's the significance?