いstrengthens a question, assertion, etc.

上司(じょうし)：「やあ、元気(げんき)かい？」部下(ぶか)：「はい！」

Boss: "Hey, how are you doing?" Subordinate: "Yes!" {strengthens a question, assertion, etc.}

先輩(せんぱい)：「それは何(なん)だい？」後輩(こうはい)：「これはお守(まも)りです。」

Senior: "What is that?" Junior: "This is a talisman." {strengthens a question, assertion, etc.}

レオ：「どうしたんだい？」ナナ：「彼氏(かれし)に振(ふ)られちゃったの…。」

Leo: "What's the matter?" Nana: "I've been dumped by my boyfriend..." {strengthens a question, assertion, etc.}

子供(こども)：「大変(たいへん)だい！休日(きゅうじつ)だと思(おも)いきや、今日(きょう)は学校(がっこう)がある平日(へいじつ)だった！」

Child: "Oh no! Just when I thought it was a holiday, turns out today is a weekday with school!" {strengthens a question, assertion, etc.}

お爺(じい)さん：「お湯(ゆ)加減(かげん)はどうだったかい？」お婆(ばあ)さん：「言(い)うことなしでしたよ。」

Grandfather: "How was the water temperature?" Grandmother: "There was nothing to complain about." {strengthens a question, assertion, etc.}